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Properties

Label 322.2.k.b.111.4
Level $322$
Weight $2$
Character 322.111
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 111.4
Character \(\chi\) \(=\) 322.111
Dual form 322.2.k.b.293.4

$q$-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+(-0.415415 - 0.909632i) q^{2} +(-0.0730120 - 0.248656i) q^{3} +(-0.654861 + 0.755750i) q^{4} +(0.853287 + 0.548374i) q^{5} +(-0.195855 + 0.169710i) q^{6} +(-1.00334 + 2.44812i) q^{7} +(0.959493 + 0.281733i) q^{8} +(2.46726 - 1.58561i) q^{9} +(0.144351 - 1.00398i) q^{10} +(-0.0397101 - 0.0181350i) q^{11} +(0.235735 + 0.107656i) q^{12} +(3.02501 + 0.434931i) q^{13} +(2.64369 - 0.104317i) q^{14} +(0.0740565 - 0.252213i) q^{15} +(-0.142315 - 0.989821i) q^{16} +(4.92931 + 5.68873i) q^{17} +(-2.46726 - 1.58561i) q^{18} +(-0.121167 + 0.139834i) q^{19} +(-0.973218 + 0.285763i) q^{20} +(0.681997 + 0.0707441i) q^{21} +0.0436551i q^{22} +(1.78996 - 4.44928i) q^{23} -0.259154i q^{24} +(-1.64969 - 3.61232i) q^{25} +(-0.861008 - 2.93232i) q^{26} +(-1.16198 - 1.00686i) q^{27} +(-1.19312 - 2.36145i) q^{28} +(2.19196 + 2.52966i) q^{29} +(-0.260185 + 0.0374090i) q^{30} +(0.781429 - 2.66130i) q^{31} +(-0.841254 + 0.540641i) q^{32} +(-0.00161006 + 0.0111982i) q^{33} +(3.12694 - 6.84704i) q^{34} +(-2.19863 + 1.53875i) q^{35} +(-0.417387 + 2.90299i) q^{36} +(-1.94368 - 3.02443i) q^{37} +(0.177533 + 0.0521283i) q^{38} +(-0.112714 - 0.783943i) q^{39} +(0.664228 + 0.766560i) q^{40} +(-2.41104 + 3.75165i) q^{41} +(-0.218961 - 0.649754i) q^{42} +(2.36096 + 8.04071i) q^{43} +(0.0397101 - 0.0181350i) q^{44} +2.97479 q^{45} +(-4.79078 + 0.220093i) q^{46} +0.448877i q^{47} +(-0.235735 + 0.107656i) q^{48} +(-4.98662 - 4.91260i) q^{49} +(-2.60057 + 3.00122i) q^{50} +(1.05464 - 1.64105i) q^{51} +(-2.30966 + 2.00133i) q^{52} +(3.87902 - 0.557719i) q^{53} +(-0.433169 + 1.47524i) q^{54} +(-0.0239393 - 0.0372503i) q^{55} +(-1.65241 + 2.06628i) q^{56} +(0.0436174 + 0.0199194i) q^{57} +(1.39048 - 3.04474i) q^{58} +(-11.6558 - 1.67585i) q^{59} +(0.142113 + 0.221133i) q^{60} +(2.16631 + 0.636087i) q^{61} +(-2.74542 + 0.394732i) q^{62} +(1.40627 + 7.63107i) q^{63} +(0.841254 + 0.540641i) q^{64} +(2.34270 + 2.02996i) q^{65} +(0.0108551 - 0.00318735i) q^{66} +(-0.692887 + 0.316431i) q^{67} -7.52726 q^{68} +(-1.23703 - 0.120233i) q^{69} +(2.31304 + 1.36072i) q^{70} +(4.23704 + 9.27783i) q^{71} +(2.81404 - 0.826276i) q^{72} +(-8.69175 - 7.53145i) q^{73} +(-1.94368 + 3.02443i) q^{74} +(-0.777778 + 0.673948i) q^{75} +(-0.0263322 - 0.183144i) q^{76} +(0.0842394 - 0.0790196i) q^{77} +(-0.666276 + 0.428190i) q^{78} +(-11.1625 - 1.60492i) q^{79} +(0.421357 - 0.922644i) q^{80} +(3.48951 - 7.64097i) q^{81} +(4.41420 + 0.634667i) q^{82} +(-11.1166 + 7.14418i) q^{83} +(-0.500078 + 0.469091i) q^{84} +(1.08657 + 7.55723i) q^{85} +(6.33330 - 5.48784i) q^{86} +(0.468976 - 0.729741i) q^{87} +(-0.0329923 - 0.0285880i) q^{88} +(-10.4134 + 3.05766i) q^{89} +(-1.23577 - 2.70597i) q^{90} +(-4.09988 + 6.96922i) q^{91} +(2.19036 + 4.26642i) q^{92} -0.718803 q^{93} +(0.408313 - 0.186470i) q^{94} +(-0.180072 + 0.0528739i) q^{95} +(0.195855 + 0.169710i) q^{96} +(7.55914 + 4.85796i) q^{97} +(-2.39714 + 6.57676i) q^{98} +(-0.126730 + 0.0182210i) 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 \(\left(\mathbb{Z}/322\mathbb{Z}\right)^\times\).

\(n\) \(185\) \(281\)
\(\chi(n)\) \(-1\) \(e\left(\frac{15}{22}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.415415 − 0.909632i −0.293743 − 0.643207i
\(3\) −0.0730120 − 0.248656i −0.0421535 − 0.143562i 0.935727 − 0.352724i \(-0.114745\pi\)
−0.977881 + 0.209162i \(0.932926\pi\)
\(4\) −0.654861 + 0.755750i −0.327430 + 0.377875i
\(5\) 0.853287 + 0.548374i 0.381602 + 0.245241i 0.717353 − 0.696710i \(-0.245354\pi\)
−0.335751 + 0.941951i \(0.608990\pi\)
\(6\) −0.195855 + 0.169710i −0.0799576 + 0.0692837i
\(7\) −1.00334 + 2.44812i −0.379227 + 0.925304i
\(8\) 0.959493 + 0.281733i 0.339232 + 0.0996075i
\(9\) 2.46726 − 1.58561i 0.822420 − 0.528538i
\(10\) 0.144351 − 1.00398i 0.0456477 − 0.317487i
\(11\) −0.0397101 − 0.0181350i −0.0119730 − 0.00546790i 0.409419 − 0.912346i \(-0.365731\pi\)
−0.421393 + 0.906878i \(0.638459\pi\)
\(12\) 0.235735 + 0.107656i 0.0680507 + 0.0310777i
\(13\) 3.02501 + 0.434931i 0.838987 + 0.120628i 0.548395 − 0.836219i \(-0.315239\pi\)
0.290591 + 0.956847i \(0.406148\pi\)
\(14\) 2.64369 − 0.104317i 0.706557 − 0.0278800i
\(15\) 0.0740565 − 0.252213i 0.0191213 − 0.0651212i
\(16\) −0.142315 − 0.989821i −0.0355787 − 0.247455i
\(17\) 4.92931 + 5.68873i 1.19553 + 1.37972i 0.906395 + 0.422431i \(0.138823\pi\)
0.289138 + 0.957287i \(0.406631\pi\)
\(18\) −2.46726 − 1.58561i −0.581539 − 0.373732i
\(19\) −0.121167 + 0.139834i −0.0277977 + 0.0320802i −0.769478 − 0.638673i \(-0.779484\pi\)
0.741681 + 0.670753i \(0.234029\pi\)
\(20\) −0.973218 + 0.285763i −0.217618 + 0.0638985i
\(21\) 0.681997 + 0.0707441i 0.148824 + 0.0154376i
\(22\) 0.0436551i 0.00930730i
\(23\) 1.78996 − 4.44928i 0.373232 − 0.927738i
\(24\) − 0.259154i − 0.0528995i
\(25\) −1.64969 − 3.61232i −0.329938 − 0.722464i
\(26\) −0.861008 − 2.93232i −0.168857 − 0.575076i
\(27\) −1.16198 − 1.00686i −0.223623 − 0.193770i
\(28\) −1.19312 − 2.36145i −0.225479 − 0.446273i
\(29\) 2.19196 + 2.52966i 0.407037 + 0.469746i 0.921844 − 0.387560i \(-0.126682\pi\)
−0.514807 + 0.857306i \(0.672137\pi\)
\(30\) −0.260185 + 0.0374090i −0.0475031 + 0.00682992i
\(31\) 0.781429 − 2.66130i 0.140349 − 0.477984i −0.859078 − 0.511845i \(-0.828962\pi\)
0.999427 + 0.0338611i \(0.0107804\pi\)
\(32\) −0.841254 + 0.540641i −0.148714 + 0.0955727i
\(33\) −0.00161006 + 0.0111982i −0.000280276 + 0.00194936i
\(34\) 3.12694 − 6.84704i 0.536265 − 1.17426i
\(35\) −2.19863 + 1.53875i −0.371636 + 0.260096i
\(36\) −0.417387 + 2.90299i −0.0695644 + 0.483831i
\(37\) −1.94368 − 3.02443i −0.319539 − 0.497213i 0.643911 − 0.765101i \(-0.277311\pi\)
−0.963450 + 0.267888i \(0.913674\pi\)
\(38\) 0.177533 + 0.0521283i 0.0287996 + 0.00845632i
\(39\) −0.112714 − 0.783943i −0.0180487 − 0.125531i
\(40\) 0.664228 + 0.766560i 0.105024 + 0.121204i
\(41\) −2.41104 + 3.75165i −0.376541 + 0.585910i −0.976867 − 0.213846i \(-0.931401\pi\)
0.600326 + 0.799755i \(0.295037\pi\)
\(42\) −0.218961 − 0.649754i −0.0337864 − 0.100259i
\(43\) 2.36096 + 8.04071i 0.360044 + 1.22620i 0.918085 + 0.396383i \(0.129735\pi\)
−0.558041 + 0.829813i \(0.688447\pi\)
\(44\) 0.0397101 − 0.0181350i 0.00598652 − 0.00273395i
\(45\) 2.97479 0.443456
\(46\) −4.79078 + 0.220093i −0.706362 + 0.0324509i
\(47\) 0.448877i 0.0654754i 0.999464 + 0.0327377i \(0.0104226\pi\)
−0.999464 + 0.0327377i \(0.989577\pi\)
\(48\) −0.235735 + 0.107656i −0.0340254 + 0.0155389i
\(49\) −4.98662 − 4.91260i −0.712374 − 0.701800i
\(50\) −2.60057 + 3.00122i −0.367777 + 0.424437i
\(51\) 1.05464 − 1.64105i 0.147679 − 0.229793i
\(52\) −2.30966 + 2.00133i −0.320292 + 0.277535i
\(53\) 3.87902 − 0.557719i 0.532825 − 0.0766086i 0.129349 − 0.991599i \(-0.458711\pi\)
0.403475 + 0.914991i \(0.367802\pi\)
\(54\) −0.433169 + 1.47524i −0.0589468 + 0.200754i
\(55\) −0.0239393 − 0.0372503i −0.00322798 − 0.00502283i
\(56\) −1.65241 + 2.06628i −0.220813 + 0.276119i
\(57\) 0.0436174 + 0.0199194i 0.00577726 + 0.00263839i
\(58\) 1.39048 − 3.04474i 0.182580 − 0.399794i
\(59\) −11.6558 − 1.67585i −1.51745 − 0.218176i −0.667329 − 0.744763i \(-0.732563\pi\)
−0.850122 + 0.526587i \(0.823472\pi\)
\(60\) 0.142113 + 0.221133i 0.0183467 + 0.0285481i
\(61\) 2.16631 + 0.636087i 0.277368 + 0.0814426i 0.417458 − 0.908696i \(-0.362921\pi\)
−0.140090 + 0.990139i \(0.544739\pi\)
\(62\) −2.74542 + 0.394732i −0.348669 + 0.0501310i
\(63\) 1.40627 + 7.63107i 0.177174 + 0.961424i
\(64\) 0.841254 + 0.540641i 0.105157 + 0.0675801i
\(65\) 2.34270 + 2.02996i 0.290576 + 0.251785i
\(66\) 0.0108551 − 0.00318735i 0.00133617 − 0.000392335i
\(67\) −0.692887 + 0.316431i −0.0846497 + 0.0386582i −0.457290 − 0.889318i \(-0.651180\pi\)
0.372640 + 0.927976i \(0.378453\pi\)
\(68\) −7.52726 −0.912815
\(69\) −1.23703 − 0.120233i −0.148921 − 0.0144744i
\(70\) 2.31304 + 1.36072i 0.276461 + 0.162637i
\(71\) 4.23704 + 9.27783i 0.502844 + 1.10108i 0.975534 + 0.219847i \(0.0705558\pi\)
−0.472690 + 0.881229i \(0.656717\pi\)
\(72\) 2.81404 − 0.826276i 0.331638 − 0.0973776i
\(73\) −8.69175 − 7.53145i −1.01729 − 0.881489i −0.0243049 − 0.999705i \(-0.507737\pi\)
−0.992988 + 0.118215i \(0.962283\pi\)
\(74\) −1.94368 + 3.02443i −0.225948 + 0.351583i
\(75\) −0.777778 + 0.673948i −0.0898101 + 0.0778209i
\(76\) −0.0263322 − 0.183144i −0.00302050 − 0.0210081i
\(77\) 0.0842394 − 0.0790196i 0.00959997 − 0.00900512i
\(78\) −0.666276 + 0.428190i −0.0754409 + 0.0484829i
\(79\) −11.1625 − 1.60492i −1.25587 − 0.180567i −0.517926 − 0.855425i \(-0.673296\pi\)
−0.737948 + 0.674858i \(0.764205\pi\)
\(80\) 0.421357 − 0.922644i 0.0471092 − 0.103155i
\(81\) 3.48951 − 7.64097i 0.387724 − 0.848996i
\(82\) 4.41420 + 0.634667i 0.487467 + 0.0700872i
\(83\) −11.1166 + 7.14418i −1.22020 + 0.784176i −0.982337 − 0.187123i \(-0.940084\pi\)
−0.237864 + 0.971298i \(0.576447\pi\)
\(84\) −0.500078 + 0.469091i −0.0545630 + 0.0511821i
\(85\) 1.08657 + 7.55723i 0.117855 + 0.819696i
\(86\) 6.33330 − 5.48784i 0.682938 − 0.591769i
\(87\) 0.468976 − 0.729741i 0.0502795 − 0.0782364i
\(88\) −0.0329923 − 0.0285880i −0.00351699 − 0.00304749i
\(89\) −10.4134 + 3.05766i −1.10382 + 0.324111i −0.782369 − 0.622815i \(-0.785989\pi\)
−0.321451 + 0.946926i \(0.604171\pi\)
\(90\) −1.23577 − 2.70597i −0.130262 − 0.285234i
\(91\) −4.09988 + 6.96922i −0.429784 + 0.730572i
\(92\) 2.19036 + 4.26642i 0.228361 + 0.444805i
\(93\) −0.718803 −0.0745364
\(94\) 0.408313 − 0.186470i 0.0421142 − 0.0192329i
\(95\) −0.180072 + 0.0528739i −0.0184750 + 0.00542475i
\(96\) 0.195855 + 0.169710i 0.0199894 + 0.0173209i
\(97\) 7.55914 + 4.85796i 0.767514 + 0.493252i 0.864869 − 0.501998i \(-0.167402\pi\)
−0.0973543 + 0.995250i \(0.531038\pi\)
\(98\) −2.39714 + 6.57676i −0.242148 + 0.664353i
\(99\) −0.126730 + 0.0182210i −0.0127369 + 0.00183128i
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.111.4 ✓ 80
7.6 odd 2 inner 322.2.k.b.111.5 yes 80
23.17 odd 22 inner 322.2.k.b.293.5 yes 80
161.132 even 22 inner 322.2.k.b.293.4 yes 80
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
322.2.k.b.111.4 ✓ 80 1.1 even 1 trivial
322.2.k.b.111.5 yes 80 7.6 odd 2 inner
322.2.k.b.293.4 yes 80 161.132 even 22 inner
322.2.k.b.293.5 yes 80 23.17 odd 22 inner