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Properties

Label 322.2.k.b.97.3
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 97.3
Character \(\chi\) \(=\) 322.97
Dual form 322.2.k.b.83.3

$q$-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} +(2.48964 - 0.895380i) 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} +(2.64105 - 0.157699i) 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} +(-3.42090 - 1.32196i) 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} +(2.57850 + 0.592758i) 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} +(0.856378 + 4.17234i) 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} +(-2.90989 - 2.23219i) 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} +(5.39659 - 4.45834i) 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} +(2.30705 + 1.29519i) 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} +(-1.33803 - 2.52043i) 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} +(4.59264 + 6.28785i) 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} +(-2.16314 - 2.96158i) 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} +(6.43405 - 2.75736i) 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 0.297733 − 0.954649i \(-0.403770\pi\)
−0.902561 + 0.430563i \(0.858315\pi\)
\(4\) 0.841254 + 0.540641i 0.420627 + 0.270320i
\(5\) −0.229109 + 1.59349i −0.102461 + 0.712629i 0.872234 + 0.489088i \(0.162670\pi\)
−0.974695 + 0.223540i \(0.928239\pi\)
\(6\) −0.749417 − 1.16612i −0.305948 − 0.476065i
\(7\) 2.48964 − 0.895380i 0.940994 − 0.338422i
\(8\) 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 0.927019 − 0.375015i \(-0.122362\pi\)
0.323651 + 0.946177i \(0.395090\pi\)
\(14\) 2.64105 − 0.157699i 0.705850 − 0.0421469i
\(15\) 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 0.235591 − 0.971852i \(-0.424297\pi\)
0.981896 + 0.189421i \(0.0606611\pi\)
\(18\) 0.153494 − 1.06757i 0.0361788 − 0.251629i
\(19\) −3.72964 − 2.39689i −0.855638 − 0.549885i 0.0376907 − 0.999289i \(-0.488000\pi\)
−0.893328 + 0.449405i \(0.851636\pi\)
\(20\) −1.05424 + 1.21666i −0.235736 + 0.272053i
\(21\) −3.42090 − 1.32196i −0.746502 − 0.288475i
\(22\) 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
\(28\) 2.57850 + 0.592758i 0.487290 + 0.112021i
\(29\) −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 −0.129242 − 0.991613i \(-0.541254\pi\)
0.963127 + 0.269047i \(0.0867089\pi\)
\(32\) 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
\(35\) 0.856378 + 4.17234i 0.144754 + 0.705254i
\(36\) 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 −0.530947 − 0.847405i \(-0.678164\pi\)
−0.748181 + 0.663494i \(0.769073\pi\)
\(42\) −2.90989 − 2.23219i −0.449006 − 0.344435i
\(43\) −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 \(0.134200\pi\)
−0.912435 + 0.409222i \(0.865800\pi\)
\(48\) 0.390527 − 1.33001i 0.0563678 − 0.191971i
\(49\) 5.39659 − 4.45834i 0.770941 − 0.636906i
\(50\) 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
\(56\) 2.30705 + 1.29519i 0.308292 + 0.173077i
\(57\) 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 −0.318916 − 0.947783i \(-0.603319\pi\)
−0.925132 + 0.379645i \(0.876046\pi\)
\(60\) 2.20883 − 0.317582i 0.285159 − 0.0409997i
\(61\) 3.34505 + 3.86040i 0.428290 + 0.494273i 0.928345 − 0.371721i \(-0.121232\pi\)
−0.500054 + 0.865994i \(0.666687\pi\)
\(62\) 5.58824 − 2.55206i 0.709707 − 0.324112i
\(63\) −1.33803 − 2.52043i −0.168576 − 0.317545i
\(64\) −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 \(0.510632\pi\)
−0.895252 + 0.445561i \(0.853004\pi\)
\(74\) −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
\(77\) 4.59264 + 6.28785i 0.523380 + 0.716566i
\(78\) −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 \(-0.926108\pi\)
0.868940 − 0.494917i \(-0.164801\pi\)
\(84\) −2.16314 − 2.96158i −0.236018 − 0.323135i
\(85\) 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 −0.849279 − 0.527944i \(-0.822963\pi\)
0.643435 + 0.765501i \(0.277509\pi\)
\(90\) 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 \(-0.564445\pi\)
0.468912 − 0.883245i \(-0.344646\pi\)
\(98\) 6.43405 − 2.75736i 0.649937 − 0.278535i
\(99\) 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.97.3 yes 80
7.6 odd 2 inner 322.2.k.b.97.6 yes 80
23.14 odd 22 inner 322.2.k.b.83.6 yes 80
161.83 even 22 inner 322.2.k.b.83.3 ✓ 80
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
322.2.k.b.83.3 ✓ 80 161.83 even 22 inner
322.2.k.b.83.6 yes 80 23.14 odd 22 inner
322.2.k.b.97.3 yes 80 1.1 even 1 trivial
322.2.k.b.97.6 yes 80 7.6 odd 2 inner