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Properties

Label 322.2.o.b.171.1
Level $322$
Weight $2$
Character 322.171
Analytic conductor $2.571$
Analytic rank $0$
Dimension $160$
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(5,322)] 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("322.5"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(322, base_ring=CyclotomicField(66)) chi = DirichletCharacter(H, H._module([55, 3])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 322 = 2 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 322.o (of order \(66\), degree \(20\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [160,8] 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: \(160\)
Relative dimension: \(8\) over \(\Q(\zeta_{66})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{66}]$

Embedding invariants

Embedding label 171.1
Character \(\chi\) \(=\) 322.171
Dual form 322.2.o.b.145.1

$q$-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+(-0.327068 + 0.945001i) q^{2} +(-1.48702 - 2.88441i) q^{3} +(-0.786053 - 0.618159i) q^{4} +(1.89904 - 0.181336i) q^{5} +(3.21212 - 0.461834i) q^{6} +(2.61415 + 0.407702i) q^{7} +(0.841254 - 0.540641i) q^{8} +(-4.36842 + 6.13459i) q^{9} +(-0.449752 + 1.85390i) q^{10} +(3.30065 - 1.14237i) q^{11} +(-0.614149 + 3.18651i) q^{12} +(-1.65923 - 5.65083i) q^{13} +(-1.24028 + 2.33703i) q^{14} +(-3.34695 - 5.20796i) q^{15} +(0.235759 + 0.971812i) q^{16} +(-4.27817 - 1.71272i) q^{17} +(-4.36842 - 6.13459i) q^{18} +(3.84010 - 1.53734i) q^{19} +(-1.60484 - 1.03137i) q^{20} +(-2.71130 - 8.14653i) q^{21} +3.49275i q^{22} +(-4.79557 - 0.0501303i) q^{23} +(-2.81039 - 1.62258i) q^{24} +(-1.33617 + 0.257526i) q^{25} +(5.88272 + 0.280228i) q^{26} +(14.5542 + 2.09258i) q^{27} +(-1.80284 - 1.93644i) q^{28} +(-0.382802 - 2.66244i) q^{29} +(6.01620 - 1.45952i) q^{30} +(2.61866 - 0.124742i) q^{31} +(-0.995472 - 0.0950560i) q^{32} +(-8.20316 - 7.82170i) q^{33} +(3.01777 - 3.48270i) q^{34} +(5.03831 + 0.300202i) q^{35} +(7.22597 - 2.12174i) q^{36} +(-0.522484 - 0.372059i) q^{37} +(0.196818 + 4.13172i) q^{38} +(-13.8320 + 13.1888i) q^{39} +(1.49954 - 1.17925i) q^{40} +(4.39045 + 2.00505i) q^{41} +(8.58526 + 0.102287i) q^{42} +(-1.69763 + 2.64157i) q^{43} +(-3.30065 - 1.14237i) q^{44} +(-7.18339 + 12.4420i) q^{45} +(1.61585 - 4.51542i) q^{46} +(-2.04626 + 1.18141i) q^{47} +(2.45252 - 2.12512i) q^{48} +(6.66756 + 2.13159i) q^{49} +(0.193657 - 1.34691i) q^{50} +(1.42152 + 14.8868i) q^{51} +(-2.18887 + 5.46752i) q^{52} +(5.72863 + 6.00801i) q^{53} +(-6.73771 + 13.0693i) q^{54} +(6.06091 - 2.76793i) q^{55} +(2.41958 - 1.07034i) q^{56} +(-10.1446 - 8.79036i) q^{57} +(2.64121 + 0.509052i) q^{58} +(4.73159 + 1.14787i) q^{59} +(-0.588464 + 6.16268i) q^{60} +(-7.86870 - 4.05659i) q^{61} +(-0.738598 + 2.51543i) q^{62} +(-13.9208 + 14.2557i) q^{63} +(0.415415 - 0.909632i) q^{64} +(-4.17565 - 10.4303i) q^{65} +(10.0745 - 5.19377i) q^{66} +(-2.05314 - 10.6527i) q^{67} +(2.30413 + 3.99088i) q^{68} +(6.98649 + 13.9069i) q^{69} +(-1.93156 + 4.66302i) q^{70} +(8.85619 + 10.2206i) q^{71} +(-0.358341 + 7.52250i) q^{72} +(3.70756 - 4.71455i) q^{73} +(0.522484 - 0.372059i) q^{74} +(2.72972 + 3.47112i) q^{75} +(-3.96885 - 1.16536i) q^{76} +(9.09413 - 1.64063i) q^{77} +(-7.93940 - 17.3849i) q^{78} +(8.91187 - 9.34650i) q^{79} +(0.623941 + 1.80276i) q^{80} +(-8.21702 - 23.7415i) q^{81} +(-3.33075 + 3.49319i) q^{82} +(1.27408 + 2.78985i) q^{83} +(-2.90463 + 8.07962i) q^{84} +(-8.43499 - 2.47674i) q^{85} +(-1.94104 - 2.46824i) q^{86} +(-7.11034 + 5.06325i) q^{87} +(2.15907 - 2.74548i) q^{88} +(-0.0232298 + 0.487653i) q^{89} +(-9.40824 - 10.8577i) q^{90} +(-2.03363 - 15.4486i) q^{91} +(3.73858 + 3.00383i) q^{92} +(-4.25379 - 7.36778i) q^{93} +(-0.447165 - 2.32011i) q^{94} +(7.01373 - 3.61583i) q^{95} +(1.20610 + 3.01270i) q^{96} +(-3.79972 + 8.32023i) q^{97} +(-4.19510 + 5.60367i) q^{98} +(-7.41068 + 25.2385i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 160 q + 8 q^{2} - 6 q^{3} + 8 q^{4} + 11 q^{7} - 16 q^{8} + 12 q^{9} - 27 q^{12} - 11 q^{14} + 8 q^{16} + 66 q^{17} + 12 q^{18} - 66 q^{21} - 18 q^{23} - 6 q^{24} - 2 q^{25} + 6 q^{26} - 22 q^{28} - 16 q^{29}+ \cdots - 220 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(185\) \(281\)
\(\chi(n)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{3}{22}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.327068 + 0.945001i −0.231272 + 0.668216i
\(3\) −1.48702 − 2.88441i −0.858529 − 1.66531i −0.740248 − 0.672334i \(-0.765292\pi\)
−0.118281 − 0.992980i \(-0.537738\pi\)
\(4\) −0.786053 − 0.618159i −0.393027 − 0.309079i
\(5\) 1.89904 − 0.181336i 0.849277 − 0.0810961i 0.338664 − 0.940907i \(-0.390025\pi\)
0.510612 + 0.859811i \(0.329419\pi\)
\(6\) 3.21212 − 0.461834i 1.31134 − 0.188543i
\(7\) 2.61415 + 0.407702i 0.988056 + 0.154097i
\(8\) 0.841254 − 0.540641i 0.297428 − 0.191145i
\(9\) −4.36842 + 6.13459i −1.45614 + 2.04486i
\(10\) −0.449752 + 1.85390i −0.142224 + 0.586256i
\(11\) 3.30065 − 1.14237i 0.995183 − 0.344436i 0.219537 − 0.975604i \(-0.429545\pi\)
0.775646 + 0.631168i \(0.217424\pi\)
\(12\) −0.614149 + 3.18651i −0.177290 + 0.919866i
\(13\) −1.65923 − 5.65083i −0.460188 − 1.56726i −0.783766 − 0.621056i \(-0.786704\pi\)
0.323577 − 0.946202i \(-0.395114\pi\)
\(14\) −1.24028 + 2.33703i −0.331480 + 0.624597i
\(15\) −3.34695 − 5.20796i −0.864179 − 1.34469i
\(16\) 0.235759 + 0.971812i 0.0589397 + 0.242953i
\(17\) −4.27817 − 1.71272i −1.03761 − 0.415396i −0.210592 − 0.977574i \(-0.567539\pi\)
−0.827016 + 0.562178i \(0.809963\pi\)
\(18\) −4.36842 − 6.13459i −1.02965 − 1.44594i
\(19\) 3.84010 − 1.53734i 0.880980 − 0.352691i 0.113328 − 0.993558i \(-0.463849\pi\)
0.767652 + 0.640867i \(0.221425\pi\)
\(20\) −1.60484 − 1.03137i −0.358853 − 0.230621i
\(21\) −2.71130 − 8.14653i −0.591655 − 1.77772i
\(22\) 3.49275i 0.744656i
\(23\) −4.79557 − 0.0501303i −0.999945 − 0.0104529i
\(24\) −2.81039 − 1.62258i −0.573668 − 0.331207i
\(25\) −1.33617 + 0.257526i −0.267234 + 0.0515052i
\(26\) 5.88272 + 0.280228i 1.15370 + 0.0549573i
\(27\) 14.5542 + 2.09258i 2.80096 + 0.402717i
\(28\) −1.80284 − 1.93644i −0.340704 − 0.365952i
\(29\) −0.382802 − 2.66244i −0.0710845 − 0.494403i −0.993998 − 0.109397i \(-0.965108\pi\)
0.922914 − 0.385007i \(-0.125801\pi\)
\(30\) 6.01620 − 1.45952i 1.09840 − 0.266470i
\(31\) 2.61866 − 0.124742i 0.470325 − 0.0224043i 0.188918 − 0.981993i \(-0.439502\pi\)
0.281407 + 0.959589i \(0.409199\pi\)
\(32\) −0.995472 − 0.0950560i −0.175976 − 0.0168037i
\(33\) −8.20316 − 7.82170i −1.42799 − 1.36158i
\(34\) 3.01777 − 3.48270i 0.517544 − 0.597278i
\(35\) 5.03831 + 0.300202i 0.851629 + 0.0507434i
\(36\) 7.22597 − 2.12174i 1.20433 − 0.353623i
\(37\) −0.522484 − 0.372059i −0.0858958 − 0.0611661i 0.536302 − 0.844026i \(-0.319821\pi\)
−0.622198 + 0.782860i \(0.713760\pi\)
\(38\) 0.196818 + 4.13172i 0.0319281 + 0.670253i
\(39\) −13.8320 + 13.1888i −2.21489 + 2.11189i
\(40\) 1.49954 − 1.17925i 0.237098 − 0.186456i
\(41\) 4.39045 + 2.00505i 0.685673 + 0.313136i 0.727617 − 0.685984i \(-0.240628\pi\)
−0.0419443 + 0.999120i \(0.513355\pi\)
\(42\) 8.58526 + 0.102287i 1.32473 + 0.0157832i
\(43\) −1.69763 + 2.64157i −0.258887 + 0.402835i −0.946229 − 0.323498i \(-0.895141\pi\)
0.687342 + 0.726334i \(0.258777\pi\)
\(44\) −3.30065 − 1.14237i −0.497591 − 0.172218i
\(45\) −7.18339 + 12.4420i −1.07084 + 1.85474i
\(46\) 1.61585 − 4.51542i 0.238244 − 0.665763i
\(47\) −2.04626 + 1.18141i −0.298477 + 0.172326i −0.641759 − 0.766907i \(-0.721795\pi\)
0.343281 + 0.939233i \(0.388462\pi\)
\(48\) 2.45252 − 2.12512i 0.353991 − 0.306735i
\(49\) 6.66756 + 2.13159i 0.952508 + 0.304513i
\(50\) 0.193657 − 1.34691i 0.0273872 − 0.190482i
\(51\) 1.42152 + 14.8868i 0.199053 + 2.08457i
\(52\) −2.18887 + 5.46752i −0.303541 + 0.758209i
\(53\) 5.72863 + 6.00801i 0.786887 + 0.825264i 0.988024 − 0.154300i \(-0.0493121\pi\)
−0.201137 + 0.979563i \(0.564464\pi\)
\(54\) −6.73771 + 13.0693i −0.916886 + 1.77851i
\(55\) 6.06091 − 2.76793i 0.817253 − 0.373227i
\(56\) 2.41958 − 1.07034i 0.323330 − 0.143030i
\(57\) −10.1446 − 8.79036i −1.34369 − 1.16431i
\(58\) 2.64121 + 0.509052i 0.346808 + 0.0668418i
\(59\) 4.73159 + 1.14787i 0.616001 + 0.149440i 0.531605 − 0.846992i \(-0.321589\pi\)
0.0843955 + 0.996432i \(0.473104\pi\)
\(60\) −0.588464 + 6.16268i −0.0759704 + 0.795598i
\(61\) −7.86870 − 4.05659i −1.00748 − 0.519394i −0.126211 − 0.992003i \(-0.540282\pi\)
−0.881272 + 0.472610i \(0.843312\pi\)
\(62\) −0.738598 + 2.51543i −0.0938020 + 0.319460i
\(63\) −13.9208 + 14.2557i −1.75386 + 1.79605i
\(64\) 0.415415 − 0.909632i 0.0519269 − 0.113704i
\(65\) −4.17565 − 10.4303i −0.517926 − 1.29372i
\(66\) 10.0745 − 5.19377i 1.24009 − 0.639309i
\(67\) −2.05314 − 10.6527i −0.250831 − 1.30143i −0.860773 − 0.508989i \(-0.830019\pi\)
0.609942 − 0.792446i \(-0.291193\pi\)
\(68\) 2.30413 + 3.99088i 0.279417 + 0.483965i
\(69\) 6.98649 + 13.9069i 0.841075 + 1.67420i
\(70\) −1.93156 + 4.66302i −0.230866 + 0.557337i
\(71\) 8.85619 + 10.2206i 1.05104 + 1.21296i 0.976449 + 0.215746i \(0.0692185\pi\)
0.0745873 + 0.997214i \(0.476236\pi\)
\(72\) −0.358341 + 7.52250i −0.0422308 + 0.886535i
\(73\) 3.70756 − 4.71455i 0.433937 − 0.551796i −0.519075 − 0.854729i \(-0.673723\pi\)
0.953012 + 0.302933i \(0.0979657\pi\)
\(74\) 0.522484 − 0.372059i 0.0607375 − 0.0432510i
\(75\) 2.72972 + 3.47112i 0.315201 + 0.400810i
\(76\) −3.96885 − 1.16536i −0.455258 − 0.133676i
\(77\) 9.09413 − 1.64063i 1.03637 − 0.186968i
\(78\) −7.93940 − 17.3849i −0.898960 − 1.96845i
\(79\) 8.91187 − 9.34650i 1.00266 − 1.05156i 0.00399409 − 0.999992i \(-0.498729\pi\)
0.998669 − 0.0515711i \(-0.0164229\pi\)
\(80\) 0.623941 + 1.80276i 0.0697587 + 0.201554i
\(81\) −8.21702 − 23.7415i −0.913002 − 2.63795i
\(82\) −3.33075 + 3.49319i −0.367820 + 0.385758i
\(83\) 1.27408 + 2.78985i 0.139849 + 0.306226i 0.966577 − 0.256375i \(-0.0825281\pi\)
−0.826729 + 0.562601i \(0.809801\pi\)
\(84\) −2.90463 + 8.07962i −0.316921 + 0.881559i
\(85\) −8.43499 − 2.47674i −0.914904 − 0.268640i
\(86\) −1.94104 − 2.46824i −0.209308 − 0.266157i
\(87\) −7.11034 + 5.06325i −0.762309 + 0.542838i
\(88\) 2.15907 − 2.74548i 0.230158 − 0.292670i
\(89\) −0.0232298 + 0.487653i −0.00246235 + 0.0516911i −0.999764 − 0.0217204i \(-0.993086\pi\)
0.997302 + 0.0734116i \(0.0233887\pi\)
\(90\) −9.40824 − 10.8577i −0.991715 − 1.14450i
\(91\) −2.03363 − 15.4486i −0.213182 − 1.61945i
\(92\) 3.73858 + 3.00383i 0.389774 + 0.313171i
\(93\) −4.25379 − 7.36778i −0.441098 − 0.764004i
\(94\) −0.447165 − 2.32011i −0.0461216 − 0.239301i
\(95\) 7.01373 − 3.61583i 0.719594 − 0.370976i
\(96\) 1.20610 + 3.01270i 0.123097 + 0.307482i
\(97\) −3.79972 + 8.32023i −0.385803 + 0.844792i 0.612711 + 0.790307i \(0.290079\pi\)
−0.998515 + 0.0544848i \(0.982648\pi\)
\(98\) −4.19510 + 5.60367i −0.423769 + 0.566057i
\(99\) −7.41068 + 25.2385i −0.744802 + 2.53656i
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.o.b.171.1 yes 160
7.5 odd 6 inner 322.2.o.b.33.1 ✓ 160
23.7 odd 22 inner 322.2.o.b.283.1 yes 160
161.145 even 66 inner 322.2.o.b.145.1 yes 160
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
322.2.o.b.33.1 ✓ 160 7.5 odd 6 inner
322.2.o.b.145.1 yes 160 161.145 even 66 inner
322.2.o.b.171.1 yes 160 1.1 even 1 trivial
322.2.o.b.283.1 yes 160 23.7 odd 22 inner