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Properties

Label 322.2.k.b.97.6
Level $322$
Weight $2$
Character 322.97
Analytic conductor $2.571$
Analytic rank $0$
Dimension $80$
Inner twists $4$

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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] = [322,2,Mod(83,322)] 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("322.83"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(322, base_ring=CyclotomicField(22)) chi = DirichletCharacter(H, H._module([11, 21])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 322 = 2 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 322.k (of order \(22\), degree \(10\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [80,8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(2.57118294509\)
Analytic rank: \(0\)
Dimension: \(80\)
Relative dimension: \(8\) over \(\Q(\zeta_{22})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{22}]$

Embedding invariants

Embedding label 97.6
Character \(\chi\) \(=\) 322.97
Dual form 322.2.k.b.83.6

$q$-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+(0.959493 + 0.281733i) q^{2} +(1.04759 + 0.907744i) q^{3} +(0.841254 + 0.540641i) q^{4} +(0.229109 - 1.59349i) q^{5} +(0.749417 + 1.16612i) q^{6} +(2.30705 - 1.29519i) q^{7} +(0.654861 + 0.755750i) q^{8} +(-0.153494 - 1.06757i) q^{9} +(0.668765 - 1.46439i) q^{10} +(0.829143 + 2.82380i) q^{11} +(0.390527 + 1.33001i) q^{12} +(-4.50935 - 2.05935i) q^{13} +(2.57850 - 0.592758i) q^{14} +(1.68649 - 1.46135i) q^{15} +(0.415415 + 0.909632i) q^{16} +(-5.01983 + 3.22605i) q^{17} +(0.153494 - 1.06757i) q^{18} +(3.72964 + 2.39689i) q^{19} +(1.05424 - 1.21666i) q^{20} +(3.59255 + 0.737376i) q^{21} +2.94302i q^{22} +(-4.66523 + 1.11159i) q^{23} +1.38616i q^{24} +(2.31076 + 0.678500i) q^{25} +(-3.74651 - 3.24637i) q^{26} +(3.05653 - 4.75606i) q^{27} +(2.64105 + 0.157699i) q^{28} +(-2.58184 + 1.65925i) q^{29} +(2.02989 - 0.927018i) q^{30} +(-4.64288 + 4.02308i) q^{31} +(0.142315 + 0.989821i) q^{32} +(-1.69469 + 3.71084i) q^{33} +(-5.72537 + 1.68112i) q^{34} +(-1.53531 - 3.97299i) q^{35} +(0.448046 - 0.981084i) q^{36} +(-4.07187 + 0.585446i) q^{37} +(2.90328 + 3.35056i) q^{38} +(-2.85460 - 6.25070i) q^{39} +(1.35431 - 0.870362i) q^{40} +(8.19041 + 1.17760i) q^{41} +(3.23929 + 1.71965i) q^{42} +(-6.07741 - 5.26611i) q^{43} +(-0.829143 + 2.82380i) q^{44} -1.73633 q^{45} +(-4.78943 - 0.247787i) q^{46} -5.61097i q^{47} +(-0.390527 + 1.33001i) q^{48} +(3.64495 - 5.97615i) q^{49} +(2.02600 + 1.30203i) q^{50} +(-8.18716 - 1.17714i) q^{51} +(-2.68014 - 4.17038i) q^{52} +(-2.84880 + 1.30100i) q^{53} +(4.27266 - 3.70228i) q^{54} +(4.68965 - 0.674270i) q^{55} +(2.48964 + 0.895380i) q^{56} +(1.73138 + 5.89652i) q^{57} +(-2.94472 + 0.864647i) q^{58} +(9.55571 + 4.36395i) q^{59} +(2.20883 - 0.317582i) q^{60} +(-3.34505 - 3.86040i) q^{61} +(-5.58824 + 2.55206i) q^{62} +(-1.73683 - 2.26414i) q^{63} +(-0.142315 + 0.989821i) q^{64} +(-4.31468 + 6.71378i) q^{65} +(-2.67150 + 3.08308i) q^{66} +(1.61584 - 5.50306i) q^{67} -5.96708 q^{68} +(-5.89630 - 3.07034i) q^{69} +(-0.353795 - 4.24460i) q^{70} +(3.06329 + 0.899462i) q^{71} +(0.706300 - 0.815114i) q^{72} +(7.93436 - 12.3461i) q^{73} +(-4.07187 - 0.585446i) q^{74} +(1.80483 + 2.80837i) q^{75} +(1.84171 + 4.03279i) q^{76} +(5.57024 + 5.44075i) q^{77} +(-0.977943 - 6.80174i) q^{78} +(-12.5651 - 5.73828i) q^{79} +(1.54466 - 0.453553i) q^{80} +(4.41470 - 1.29627i) q^{81} +(7.52688 + 3.43741i) q^{82} +(0.949635 + 6.60485i) q^{83} +(2.62359 + 2.56260i) q^{84} +(3.99057 + 8.73814i) q^{85} +(-4.34760 - 6.76500i) q^{86} +(-4.21088 - 0.605434i) q^{87} +(-1.59111 + 2.47582i) q^{88} +(1.94193 - 2.24111i) q^{89} +(-1.66599 - 0.489180i) q^{90} +(-13.0706 + 1.08946i) q^{91} +(-4.52561 - 1.58709i) q^{92} -8.51577 q^{93} +(1.58079 - 5.38368i) q^{94} +(4.67391 - 5.39398i) q^{95} +(-0.749417 + 1.16612i) q^{96} +(-2.63785 + 18.3466i) q^{97} +(5.18098 - 4.70717i) q^{98} +(2.88734 - 1.31861i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 80 q + 8 q^{2} - 8 q^{4} + 11 q^{7} + 8 q^{8} + 14 q^{9} - 11 q^{14} - 8 q^{16} - 14 q^{18} - 33 q^{21} + 18 q^{23} + 6 q^{25} - 11 q^{28} + 8 q^{29} + 22 q^{30} + 8 q^{32} - 33 q^{35} - 8 q^{36} - 22 q^{37}+ \cdots + 132 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/322\mathbb{Z}\right)^\times\).

\(n\) \(185\) \(281\)
\(\chi(n)\) \(-1\) \(e\left(\frac{1}{22}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.959493 + 0.281733i 0.678464 + 0.199215i
\(3\) 1.04759 + 0.907744i 0.604828 + 0.524086i 0.902561 − 0.430563i \(-0.141685\pi\)
−0.297733 + 0.954649i \(0.596230\pi\)
\(4\) 0.841254 + 0.540641i 0.420627 + 0.270320i
\(5\) 0.229109 − 1.59349i 0.102461 − 0.712629i −0.872234 − 0.489088i \(-0.837330\pi\)
0.974695 − 0.223540i \(-0.0717614\pi\)
\(6\) 0.749417 + 1.16612i 0.305948 + 0.476065i
\(7\) 2.30705 − 1.29519i 0.871983 − 0.489537i
\(8\) 0.654861 + 0.755750i 0.231528 + 0.267198i
\(9\) −0.153494 − 1.06757i −0.0511646 − 0.355857i
\(10\) 0.668765 − 1.46439i 0.211482 − 0.463081i
\(11\) 0.829143 + 2.82380i 0.249996 + 0.851408i 0.984883 + 0.173219i \(0.0554169\pi\)
−0.734887 + 0.678189i \(0.762765\pi\)
\(12\) 0.390527 + 1.33001i 0.112736 + 0.383942i
\(13\) −4.50935 − 2.05935i −1.25067 − 0.571162i −0.323651 − 0.946177i \(-0.604910\pi\)
−0.927019 + 0.375015i \(0.877638\pi\)
\(14\) 2.57850 − 0.592758i 0.689132 − 0.158421i
\(15\) 1.68649 − 1.46135i 0.435450 − 0.377319i
\(16\) 0.415415 + 0.909632i 0.103854 + 0.227408i
\(17\) −5.01983 + 3.22605i −1.21749 + 0.782431i −0.981896 − 0.189421i \(-0.939339\pi\)
−0.235591 + 0.971852i \(0.575703\pi\)
\(18\) 0.153494 − 1.06757i 0.0361788 − 0.251629i
\(19\) 3.72964 + 2.39689i 0.855638 + 0.549885i 0.893328 − 0.449405i \(-0.148364\pi\)
−0.0376907 + 0.999289i \(0.512000\pi\)
\(20\) 1.05424 − 1.21666i 0.235736 − 0.272053i
\(21\) 3.59255 + 0.737376i 0.783959 + 0.160909i
\(22\) 2.94302i 0.627453i
\(23\) −4.66523 + 1.11159i −0.972768 + 0.231782i
\(24\) 1.38616i 0.282949i
\(25\) 2.31076 + 0.678500i 0.462152 + 0.135700i
\(26\) −3.74651 − 3.24637i −0.734750 − 0.636665i
\(27\) 3.05653 − 4.75606i 0.588230 − 0.915304i
\(28\) 2.64105 + 0.157699i 0.499111 + 0.0298023i
\(29\) −2.58184 + 1.65925i −0.479435 + 0.308114i −0.757944 − 0.652319i \(-0.773796\pi\)
0.278509 + 0.960434i \(0.410160\pi\)
\(30\) 2.02989 − 0.927018i 0.370605 − 0.169250i
\(31\) −4.64288 + 4.02308i −0.833885 + 0.722566i −0.963127 − 0.269047i \(-0.913291\pi\)
0.129242 + 0.991613i \(0.458746\pi\)
\(32\) 0.142315 + 0.989821i 0.0251579 + 0.174977i
\(33\) −1.69469 + 3.71084i −0.295007 + 0.645975i
\(34\) −5.72537 + 1.68112i −0.981893 + 0.288310i
\(35\) −1.53531 − 3.97299i −0.259514 − 0.671558i
\(36\) 0.448046 − 0.981084i 0.0746743 − 0.163514i
\(37\) −4.07187 + 0.585446i −0.669411 + 0.0962468i −0.468637 − 0.883391i \(-0.655255\pi\)
−0.200774 + 0.979638i \(0.564346\pi\)
\(38\) 2.90328 + 3.35056i 0.470974 + 0.543533i
\(39\) −2.85460 − 6.25070i −0.457102 − 1.00091i
\(40\) 1.35431 − 0.870362i 0.214135 − 0.137616i
\(41\) 8.19041 + 1.17760i 1.27913 + 0.183911i 0.748181 − 0.663494i \(-0.230927\pi\)
0.530947 + 0.847405i \(0.321836\pi\)
\(42\) 3.23929 + 1.71965i 0.499833 + 0.265347i
\(43\) −6.07741 − 5.26611i −0.926796 − 0.803074i 0.0539128 − 0.998546i \(-0.482831\pi\)
−0.980709 + 0.195472i \(0.937376\pi\)
\(44\) −0.829143 + 2.82380i −0.124998 + 0.425704i
\(45\) −1.73633 −0.258836
\(46\) −4.78943 − 0.247787i −0.706162 − 0.0365342i
\(47\) − 5.61097i − 0.818443i −0.912435 − 0.409222i \(-0.865800\pi\)
0.912435 − 0.409222i \(-0.134200\pi\)
\(48\) −0.390527 + 1.33001i −0.0563678 + 0.191971i
\(49\) 3.64495 − 5.97615i 0.520707 − 0.853735i
\(50\) 2.02600 + 1.30203i 0.286520 + 0.184135i
\(51\) −8.18716 − 1.17714i −1.14643 − 0.164832i
\(52\) −2.68014 − 4.17038i −0.371669 − 0.578328i
\(53\) −2.84880 + 1.30100i −0.391313 + 0.178707i −0.601348 − 0.798987i \(-0.705369\pi\)
0.210035 + 0.977694i \(0.432642\pi\)
\(54\) 4.27266 − 3.70228i 0.581435 − 0.503817i
\(55\) 4.68965 − 0.674270i 0.632353 − 0.0909186i
\(56\) 2.48964 + 0.895380i 0.332692 + 0.119650i
\(57\) 1.73138 + 5.89652i 0.229326 + 0.781014i
\(58\) −2.94472 + 0.864647i −0.386660 + 0.113534i
\(59\) 9.55571 + 4.36395i 1.24405 + 0.568138i 0.925132 − 0.379645i \(-0.123954\pi\)
0.318916 + 0.947783i \(0.396681\pi\)
\(60\) 2.20883 − 0.317582i 0.285159 − 0.0409997i
\(61\) −3.34505 − 3.86040i −0.428290 − 0.494273i 0.500054 − 0.865994i \(-0.333313\pi\)
−0.928345 + 0.371721i \(0.878768\pi\)
\(62\) −5.58824 + 2.55206i −0.709707 + 0.324112i
\(63\) −1.73683 − 2.26414i −0.218820 − 0.285254i
\(64\) −0.142315 + 0.989821i −0.0177894 + 0.123728i
\(65\) −4.31468 + 6.71378i −0.535170 + 0.832741i
\(66\) −2.67150 + 3.08308i −0.328840 + 0.379501i
\(67\) 1.61584 − 5.50306i 0.197407 − 0.672305i −0.799978 − 0.600029i \(-0.795156\pi\)
0.997385 − 0.0722761i \(-0.0230263\pi\)
\(68\) −5.96708 −0.723615
\(69\) −5.89630 − 3.07034i −0.709831 − 0.369626i
\(70\) −0.353795 − 4.24460i −0.0422866 − 0.507327i
\(71\) 3.06329 + 0.899462i 0.363545 + 0.106746i 0.458403 − 0.888744i \(-0.348422\pi\)
−0.0948581 + 0.995491i \(0.530240\pi\)
\(72\) 0.706300 − 0.815114i 0.0832383 − 0.0960621i
\(73\) 7.93436 − 12.3461i 0.928647 − 1.44500i 0.0333955 − 0.999442i \(-0.489368\pi\)
0.895252 − 0.445561i \(-0.146996\pi\)
\(74\) −4.07187 − 0.585446i −0.473345 − 0.0680568i
\(75\) 1.80483 + 2.80837i 0.208404 + 0.324283i
\(76\) 1.84171 + 4.03279i 0.211259 + 0.462593i
\(77\) 5.57024 + 5.44075i 0.634788 + 0.620031i
\(78\) −0.977943 − 6.80174i −0.110730 − 0.770145i
\(79\) −12.5651 − 5.73828i −1.41368 − 0.645607i −0.445370 − 0.895347i \(-0.646928\pi\)
−0.968313 + 0.249740i \(0.919655\pi\)
\(80\) 1.54466 − 0.453553i 0.172698 − 0.0507088i
\(81\) 4.41470 − 1.29627i 0.490523 − 0.144030i
\(82\) 7.52688 + 3.43741i 0.831204 + 0.379598i
\(83\) 0.949635 + 6.60485i 0.104236 + 0.724977i 0.973176 + 0.230060i \(0.0738923\pi\)
−0.868940 + 0.494917i \(0.835199\pi\)
\(84\) 2.62359 + 2.56260i 0.286257 + 0.279603i
\(85\) 3.99057 + 8.73814i 0.432838 + 0.947784i
\(86\) −4.34760 − 6.76500i −0.468814 − 0.729488i
\(87\) −4.21088 − 0.605434i −0.451454 − 0.0649093i
\(88\) −1.59111 + 2.47582i −0.169613 + 0.263923i
\(89\) 1.94193 − 2.24111i 0.205844 − 0.237557i −0.643435 − 0.765501i \(-0.722491\pi\)
0.849279 + 0.527944i \(0.177037\pi\)
\(90\) −1.66599 − 0.489180i −0.175611 − 0.0515641i
\(91\) −13.0706 + 1.08946i −1.37017 + 0.114206i
\(92\) −4.52561 − 1.58709i −0.471828 − 0.165465i
\(93\) −8.51577 −0.883044
\(94\) 1.58079 − 5.38368i 0.163046 − 0.555284i
\(95\) 4.67391 − 5.39398i 0.479533 − 0.553410i
\(96\) −0.749417 + 1.16612i −0.0764870 + 0.119016i
\(97\) −2.63785 + 18.3466i −0.267833 + 1.86282i 0.201079 + 0.979575i \(0.435555\pi\)
−0.468912 + 0.883245i \(0.655354\pi\)
\(98\) 5.18098 − 4.70717i 0.523358 − 0.475496i
\(99\) 2.88734 − 1.31861i 0.290189 − 0.132525i
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 322.2.k.b.97.6 yes 80
7.6 odd 2 inner 322.2.k.b.97.3 yes 80
23.14 odd 22 inner 322.2.k.b.83.3 ✓ 80
161.83 even 22 inner 322.2.k.b.83.6 yes 80
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
322.2.k.b.83.3 ✓ 80 23.14 odd 22 inner
322.2.k.b.83.6 yes 80 161.83 even 22 inner
322.2.k.b.97.3 yes 80 7.6 odd 2 inner
322.2.k.b.97.6 yes 80 1.1 even 1 trivial