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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 404.1
Character \(\chi\) \(=\) 567.404
Dual form 567.2.p.e.80.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-2.28676 - 1.32026i) q^{2} +(2.48619 + 4.30622i) q^{4} +(-1.25340 + 2.17095i) q^{5} +(-0.0580272 + 2.64511i) q^{7} -7.84868i q^{8} +(5.73246 - 3.30964i) q^{10} +(2.03998 - 1.17778i) q^{11} -4.73971i q^{13} +(3.62495 - 5.97214i) q^{14} +(-5.38994 + 9.33565i) q^{16} +(1.37933 + 2.38907i) q^{17} +(3.58876 + 2.07197i) q^{19} -12.4648 q^{20} -6.21992 q^{22} +(-0.200503 - 0.115761i) q^{23} +(-0.642022 - 1.11202i) q^{25} +(-6.25768 + 10.8386i) q^{26} +(-11.5347 + 6.32639i) q^{28} +7.96490i q^{29} +(-8.45130 + 4.87936i) q^{31} +(11.0567 - 6.38361i) q^{32} -7.28432i q^{34} +(-5.66969 - 3.44136i) q^{35} +(0.854274 - 1.47965i) q^{37} +(-5.47110 - 9.47622i) q^{38} +(17.0391 + 9.83753i) q^{40} -1.50422 q^{41} -5.21992 q^{43} +(10.1436 + 5.85638i) q^{44} +(0.305669 + 0.529435i) q^{46} +(-3.49901 + 6.06047i) q^{47} +(-6.99327 - 0.306977i) q^{49} +3.39056i q^{50} +(20.4102 - 11.7839i) q^{52} +(-4.48584 + 2.58990i) q^{53} +5.90492i q^{55} +(20.7607 + 0.455437i) q^{56} +(10.5158 - 18.2138i) q^{58} +(3.68092 + 6.37554i) q^{59} +(4.67975 + 2.70186i) q^{61} +25.7682 q^{62} -12.1524 q^{64} +(10.2897 + 5.94076i) q^{65} +(-2.82255 - 4.88880i) q^{67} +(-6.85857 + 11.8794i) q^{68} +(8.42173 + 15.3551i) q^{70} -9.16662i q^{71} +(-9.24552 + 5.33790i) q^{73} +(-3.90705 + 2.25574i) q^{74} +20.6053i q^{76} +(2.99699 + 5.46431i) q^{77} +(-1.49748 + 2.59371i) q^{79} +(-13.5115 - 23.4026i) q^{80} +(3.43979 + 1.98596i) q^{82} -0.946600 q^{83} -6.91541 q^{85} +(11.9367 + 6.89168i) q^{86} +(-9.24402 - 16.0111i) q^{88} +(-4.30989 + 7.46494i) q^{89} +(12.5371 + 0.275032i) q^{91} -1.15121i q^{92} +(16.0028 - 9.23925i) q^{94} +(-8.99630 + 5.19402i) q^{95} -10.8324i q^{97} +(15.5867 + 9.93494i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 32 q + 16 q^{4} - 8 q^{7} - 28 q^{16} + 24 q^{22} - 16 q^{25} - 16 q^{28} - 48 q^{31} - 4 q^{37} + 56 q^{43} + 12 q^{46} - 4 q^{49} + 48 q^{52} + 36 q^{58} + 12 q^{61} - 80 q^{64} - 20 q^{67} + 120 q^{70}+ \cdots + 72 q^{94}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.28676 − 1.32026i −1.61699 − 0.933568i −0.987694 − 0.156400i \(-0.950011\pi\)
−0.629293 − 0.777168i \(-0.716655\pi\)
\(3\) 0 0
\(4\) 2.48619 + 4.30622i 1.24310 + 2.15311i
\(5\) −1.25340 + 2.17095i −0.560537 + 0.970879i 0.436912 + 0.899504i \(0.356072\pi\)
−0.997450 + 0.0713751i \(0.977261\pi\)
\(6\) 0 0
\(7\) −0.0580272 + 2.64511i −0.0219322 + 0.999759i
\(8\) − 7.84868i − 2.77493i
\(9\) 0 0
\(10\) 5.73246 − 3.30964i 1.81276 − 1.04660i
\(11\) 2.03998 − 1.17778i 0.615076 − 0.355114i −0.159874 − 0.987137i \(-0.551109\pi\)
0.774949 + 0.632023i \(0.217775\pi\)
\(12\) 0 0
\(13\) − 4.73971i − 1.31456i −0.753646 − 0.657280i \(-0.771707\pi\)
0.753646 − 0.657280i \(-0.228293\pi\)
\(14\) 3.62495 − 5.97214i 0.968807 − 1.59612i
\(15\) 0 0
\(16\) −5.38994 + 9.33565i −1.34749 + 2.33391i
\(17\) 1.37933 + 2.38907i 0.334537 + 0.579435i 0.983396 − 0.181474i \(-0.0580868\pi\)
−0.648859 + 0.760909i \(0.724753\pi\)
\(18\) 0 0
\(19\) 3.58876 + 2.07197i 0.823318 + 0.475343i 0.851559 − 0.524258i \(-0.175657\pi\)
−0.0282413 + 0.999601i \(0.508991\pi\)
\(20\) −12.4648 −2.78721
\(21\) 0 0
\(22\) −6.21992 −1.32609
\(23\) −0.200503 − 0.115761i −0.0418078 − 0.0241378i 0.478950 − 0.877842i \(-0.341017\pi\)
−0.520758 + 0.853704i \(0.674351\pi\)
\(24\) 0 0
\(25\) −0.642022 − 1.11202i −0.128404 − 0.222403i
\(26\) −6.25768 + 10.8386i −1.22723 + 2.12563i
\(27\) 0 0
\(28\) −11.5347 + 6.32639i −2.17985 + 1.19558i
\(29\) 7.96490i 1.47904i 0.673132 + 0.739522i \(0.264948\pi\)
−0.673132 + 0.739522i \(0.735052\pi\)
\(30\) 0 0
\(31\) −8.45130 + 4.87936i −1.51790 + 0.876359i −0.518119 + 0.855308i \(0.673368\pi\)
−0.999778 + 0.0210505i \(0.993299\pi\)
\(32\) 11.0567 − 6.38361i 1.95457 − 1.12847i
\(33\) 0 0
\(34\) − 7.28432i − 1.24925i
\(35\) −5.66969 − 3.44136i −0.958352 − 0.581696i
\(36\) 0 0
\(37\) 0.854274 − 1.47965i 0.140442 − 0.243252i −0.787221 − 0.616671i \(-0.788481\pi\)
0.927663 + 0.373418i \(0.121814\pi\)
\(38\) −5.47110 − 9.47622i −0.887530 − 1.53725i
\(39\) 0 0
\(40\) 17.0391 + 9.83753i 2.69412 + 1.55545i
\(41\) −1.50422 −0.234919 −0.117460 − 0.993078i \(-0.537475\pi\)
−0.117460 + 0.993078i \(0.537475\pi\)
\(42\) 0 0
\(43\) −5.21992 −0.796031 −0.398016 − 0.917379i \(-0.630301\pi\)
−0.398016 + 0.917379i \(0.630301\pi\)
\(44\) 10.1436 + 5.85638i 1.52920 + 0.882883i
\(45\) 0 0
\(46\) 0.305669 + 0.529435i 0.0450685 + 0.0780609i
\(47\) −3.49901 + 6.06047i −0.510384 + 0.884010i 0.489544 + 0.871979i \(0.337163\pi\)
−0.999928 + 0.0120318i \(0.996170\pi\)
\(48\) 0 0
\(49\) −6.99327 − 0.306977i −0.999038 − 0.0438539i
\(50\) 3.39056i 0.479497i
\(51\) 0 0
\(52\) 20.4102 − 11.7839i 2.83039 − 1.63413i
\(53\) −4.48584 + 2.58990i −0.616178 + 0.355750i −0.775379 − 0.631496i \(-0.782441\pi\)
0.159202 + 0.987246i \(0.449108\pi\)
\(54\) 0 0
\(55\) 5.90492i 0.796219i
\(56\) 20.7607 + 0.455437i 2.77426 + 0.0608603i
\(57\) 0 0
\(58\) 10.5158 − 18.2138i 1.38079 − 2.39160i
\(59\) 3.68092 + 6.37554i 0.479215 + 0.830024i 0.999716 − 0.0238368i \(-0.00758822\pi\)
−0.520501 + 0.853861i \(0.674255\pi\)
\(60\) 0 0
\(61\) 4.67975 + 2.70186i 0.599181 + 0.345937i 0.768719 − 0.639586i \(-0.220894\pi\)
−0.169538 + 0.985524i \(0.554228\pi\)
\(62\) 25.7682 3.27256
\(63\) 0 0
\(64\) −12.1524 −1.51906
\(65\) 10.2897 + 5.94076i 1.27628 + 0.736860i
\(66\) 0 0
\(67\) −2.82255 − 4.88880i −0.344829 − 0.597262i 0.640494 − 0.767964i \(-0.278730\pi\)
−0.985323 + 0.170702i \(0.945396\pi\)
\(68\) −6.85857 + 11.8794i −0.831724 + 1.44059i
\(69\) 0 0
\(70\) 8.42173 + 15.3551i 1.00659 + 1.83528i
\(71\) − 9.16662i − 1.08788i −0.839125 − 0.543939i \(-0.816932\pi\)
0.839125 − 0.543939i \(-0.183068\pi\)
\(72\) 0 0
\(73\) −9.24552 + 5.33790i −1.08211 + 0.624754i −0.931464 − 0.363834i \(-0.881467\pi\)
−0.150642 + 0.988588i \(0.548134\pi\)
\(74\) −3.90705 + 2.25574i −0.454185 + 0.262224i
\(75\) 0 0
\(76\) 20.6053i 2.36359i
\(77\) 2.99699 + 5.46431i 0.341539 + 0.622716i
\(78\) 0 0
\(79\) −1.49748 + 2.59371i −0.168480 + 0.291815i −0.937885 − 0.346945i \(-0.887219\pi\)
0.769406 + 0.638760i \(0.220552\pi\)
\(80\) −13.5115 − 23.4026i −1.51063 − 2.61649i
\(81\) 0 0
\(82\) 3.43979 + 1.98596i 0.379861 + 0.219313i
\(83\) −0.946600 −0.103903 −0.0519514 − 0.998650i \(-0.516544\pi\)
−0.0519514 + 0.998650i \(0.516544\pi\)
\(84\) 0 0
\(85\) −6.91541 −0.750082
\(86\) 11.9367 + 6.89168i 1.28717 + 0.743149i
\(87\) 0 0
\(88\) −9.24402 − 16.0111i −0.985416 − 1.70679i
\(89\) −4.30989 + 7.46494i −0.456847 + 0.791283i −0.998792 − 0.0491313i \(-0.984355\pi\)
0.541945 + 0.840414i \(0.317688\pi\)
\(90\) 0 0
\(91\) 12.5371 + 0.275032i 1.31424 + 0.0288312i
\(92\) − 1.15121i − 0.120022i
\(93\) 0 0
\(94\) 16.0028 − 9.23925i 1.65057 − 0.952955i
\(95\) −8.99630 + 5.19402i −0.923001 + 0.532895i
\(96\) 0 0
\(97\) − 10.8324i − 1.09986i −0.835210 − 0.549931i \(-0.814654\pi\)
0.835210 − 0.549931i \(-0.185346\pi\)
\(98\) 15.5867 + 9.93494i 1.57449 + 1.00358i
\(99\) 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.404.1 yes 32
3.2 odd 2 inner 567.2.p.e.404.16 yes 32
7.3 odd 6 inner 567.2.p.e.80.16 yes 32
9.2 odd 6 567.2.i.g.215.1 32
9.4 even 3 567.2.s.g.26.16 32
9.5 odd 6 567.2.s.g.26.1 32
9.7 even 3 567.2.i.g.215.16 32
21.17 even 6 inner 567.2.p.e.80.1 ✓ 32
63.31 odd 6 567.2.i.g.269.16 32
63.38 even 6 567.2.s.g.458.16 32
63.52 odd 6 567.2.s.g.458.1 32
63.59 even 6 567.2.i.g.269.1 32
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
567.2.i.g.215.1 32 9.2 odd 6
567.2.i.g.215.16 32 9.7 even 3
567.2.i.g.269.1 32 63.59 even 6
567.2.i.g.269.16 32 63.31 odd 6
567.2.p.e.80.1 ✓ 32 21.17 even 6 inner
567.2.p.e.80.16 yes 32 7.3 odd 6 inner
567.2.p.e.404.1 yes 32 1.1 even 1 trivial
567.2.p.e.404.16 yes 32 3.2 odd 2 inner
567.2.s.g.26.1 32 9.5 odd 6
567.2.s.g.26.16 32 9.4 even 3
567.2.s.g.458.1 32 63.52 odd 6
567.2.s.g.458.16 32 63.38 even 6