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Properties

Label 323.2.bb.a.9.15
Level $323$
Weight $2$
Character 323.9
Analytic conductor $2.579$
Analytic rank $0$
Dimension $672$
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] = [323,2,Mod(9,323)] 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("323.9"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(323, base_ring=CyclotomicField(72)) chi = DirichletCharacter(H, H._module([9, 32])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 323 = 17 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 323.bb (of order \(72\), degree \(24\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(2.57916798529\)
Analytic rank: \(0\)
Dimension: \(672\)
Relative dimension: \(28\) over \(\Q(\zeta_{72})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{72}]$

Embedding invariants

Embedding label 9.15
Character \(\chi\) \(=\) 323.9
Dual form 323.2.bb.a.36.15

$q$-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+(-0.00926327 - 0.00648621i) q^{2} +(0.818709 - 0.258138i) q^{3} +(-0.683997 - 1.87927i) q^{4} +(1.65303 - 1.80397i) q^{5} +(-0.00925826 - 0.00291912i) q^{6} +(1.14584 + 0.150852i) q^{7} +(-0.0117069 + 0.0436907i) q^{8} +(-1.85381 + 1.29805i) q^{9} +(-0.0270134 + 0.00598873i) q^{10} +(-1.87540 - 1.43904i) q^{11} +(-1.04510 - 1.36201i) q^{12} +(2.78782 - 3.32239i) q^{13} +(-0.00963574 - 0.00882952i) q^{14} +(0.887681 - 1.90364i) q^{15} +(-3.06359 + 2.57066i) q^{16} +(1.84256 + 3.68849i) q^{17} +0.0255917 q^{18} +(-4.00334 - 1.72431i) q^{19} +(-4.52081 - 1.87258i) q^{20} +(0.977048 - 0.172280i) q^{21} +(0.00803836 + 0.0254944i) q^{22} +(3.40820 - 3.12304i) q^{23} +(0.00169370 + 0.0387920i) q^{24} +(-0.0860083 - 0.983079i) q^{25} +(-0.0473740 + 0.0126938i) q^{26} +(-2.75041 + 3.58440i) q^{27} +(-0.500257 - 2.25651i) q^{28} +(0.0802611 + 0.0511320i) q^{29} +(-0.0205702 + 0.0118762i) q^{30} +(4.32199 - 3.31638i) q^{31} +(0.135172 - 0.0118261i) q^{32} +(-1.90687 - 0.694046i) q^{33} +(0.00685619 - 0.0461187i) q^{34} +(2.16624 - 1.81769i) q^{35} +(3.70738 + 2.59593i) q^{36} +(9.66122 + 4.00181i) q^{37} +(0.0258998 + 0.0419392i) q^{38} +(1.42478 - 3.43971i) q^{39} +(0.0594649 + 0.0933412i) q^{40} +(-1.73196 - 0.901602i) q^{41} +(-0.0101681 - 0.00474146i) q^{42} +(0.911027 + 1.95370i) q^{43} +(-1.42158 + 4.50866i) q^{44} +(-0.722763 + 5.48993i) q^{45} +(-0.0518277 + 0.00682325i) q^{46} +(12.5155 + 2.20683i) q^{47} +(-1.84461 + 2.89545i) q^{48} +(-5.47129 - 1.46603i) q^{49} +(-0.00557974 + 0.00966439i) q^{50} +(2.46066 + 2.54417i) q^{51} +(-8.15051 - 2.96654i) q^{52} +(-3.62072 + 7.76466i) q^{53} +(0.0487270 - 0.0153636i) q^{54} +(-5.69608 + 1.00437i) q^{55} +(-0.0200051 + 0.0482965i) q^{56} +(-3.72268 - 0.378291i) q^{57} +(-0.000411827 - 0.000994239i) q^{58} +(-7.24544 + 10.3476i) q^{59} +(-4.18461 - 0.366106i) q^{60} +(-2.73901 - 2.98911i) q^{61} +(-0.0615464 + 0.00268718i) q^{62} +(-2.31997 + 1.20770i) q^{63} +(6.92555 + 3.99847i) q^{64} +(-1.38514 - 10.5212i) q^{65} +(0.0131622 + 0.0187975i) q^{66} +(1.02420 - 5.80851i) q^{67} +(5.67135 - 5.98558i) q^{68} +(1.98415 - 3.43665i) q^{69} +(-0.0318564 + 0.00278707i) q^{70} +(-2.72624 - 2.49814i) q^{71} +(-0.0350104 - 0.0961903i) q^{72} +(3.53091 + 11.1986i) q^{73} +(-0.0635379 - 0.0997345i) q^{74} +(-0.324186 - 0.782654i) q^{75} +(-0.502157 + 8.70277i) q^{76} +(-1.93181 - 1.93181i) q^{77} +(-0.0355088 + 0.0226216i) q^{78} +(0.355563 - 0.683030i) q^{79} +(-0.426830 + 9.77601i) q^{80} +(0.995542 - 2.73523i) q^{81} +(0.0101956 + 0.0195856i) q^{82} +(1.62433 + 6.06206i) q^{83} +(-0.992057 - 1.71829i) q^{84} +(9.69975 + 2.77328i) q^{85} +(0.00423304 - 0.0240068i) q^{86} +(0.0789096 + 0.0211438i) q^{87} +(0.0848279 - 0.0650907i) q^{88} +(-8.69591 + 10.3634i) q^{89} +(0.0423040 - 0.0461667i) q^{90} +(3.69558 - 3.38637i) q^{91} +(-8.20021 - 4.26876i) q^{92} +(2.68237 - 3.83082i) q^{93} +(-0.101621 - 0.101621i) q^{94} +(-9.72827 + 4.37158i) q^{95} +(0.107614 - 0.0445752i) q^{96} +(-13.4200 - 2.97515i) q^{97} +(0.0411731 + 0.0490682i) q^{98} +(5.34457 + 0.233349i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 672 q - 24 q^{2} - 24 q^{3} - 24 q^{5} - 48 q^{6} - 12 q^{7} - 12 q^{8} - 60 q^{9} - 24 q^{10} - 12 q^{12} - 24 q^{14} + 12 q^{15} - 48 q^{16} - 72 q^{17} + 48 q^{18} - 24 q^{19} - 48 q^{20} - 24 q^{23}+ \cdots - 156 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/323\mathbb{Z}\right)^\times\).

\(n\) \(20\) \(154\)
\(\chi(n)\) \(e\left(\frac{1}{8}\right)\) \(e\left(\frac{4}{9}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.00926327 − 0.00648621i −0.00655012 − 0.00458644i 0.570297 − 0.821439i \(-0.306828\pi\)
−0.576847 + 0.816852i \(0.695717\pi\)
\(3\) 0.818709 − 0.258138i 0.472682 − 0.149036i −0.0549853 − 0.998487i \(-0.517511\pi\)
0.527667 + 0.849451i \(0.323067\pi\)
\(4\) −0.683997 − 1.87927i −0.341998 − 0.939633i
\(5\) 1.65303 − 1.80397i 0.739260 − 0.806760i −0.247511 − 0.968885i \(-0.579613\pi\)
0.986770 + 0.162125i \(0.0518348\pi\)
\(6\) −0.00925826 − 0.00291912i −0.00377967 − 0.00119172i
\(7\) 1.14584 + 0.150852i 0.433086 + 0.0570168i 0.343918 − 0.939000i \(-0.388246\pi\)
0.0891680 + 0.996017i \(0.471579\pi\)
\(8\) −0.0117069 + 0.0436907i −0.00413901 + 0.0154470i
\(9\) −1.85381 + 1.29805i −0.617935 + 0.432683i
\(10\) −0.0270134 + 0.00598873i −0.00854240 + 0.00189380i
\(11\) −1.87540 − 1.43904i −0.565453 − 0.433887i 0.286067 − 0.958210i \(-0.407652\pi\)
−0.851520 + 0.524322i \(0.824319\pi\)
\(12\) −1.04510 − 1.36201i −0.301696 − 0.393177i
\(13\) 2.78782 − 3.32239i 0.773201 − 0.921465i −0.225404 − 0.974265i \(-0.572370\pi\)
0.998605 + 0.0528000i \(0.0168146\pi\)
\(14\) −0.00963574 − 0.00882952i −0.00257526 − 0.00235979i
\(15\) 0.887681 − 1.90364i 0.229198 − 0.491517i
\(16\) −3.06359 + 2.57066i −0.765898 + 0.642664i
\(17\) 1.84256 + 3.68849i 0.446887 + 0.894591i
\(18\) 0.0255917 0.00603203
\(19\) −4.00334 − 1.72431i −0.918430 − 0.395583i
\(20\) −4.52081 − 1.87258i −1.01088 − 0.418722i
\(21\) 0.977048 − 0.172280i 0.213209 − 0.0375946i
\(22\) 0.00803836 + 0.0254944i 0.00171378 + 0.00543543i
\(23\) 3.40820 − 3.12304i 0.710659 − 0.651199i −0.236623 − 0.971602i \(-0.576041\pi\)
0.947281 + 0.320403i \(0.103818\pi\)
\(24\) 0.00169370 + 0.0387920i 0.000345724 + 0.00791839i
\(25\) −0.0860083 − 0.983079i −0.0172017 − 0.196616i
\(26\) −0.0473740 + 0.0126938i −0.00929081 + 0.00248946i
\(27\) −2.75041 + 3.58440i −0.529317 + 0.689819i
\(28\) −0.500257 − 2.25651i −0.0945397 − 0.426441i
\(29\) 0.0802611 + 0.0511320i 0.0149041 + 0.00949497i 0.544731 − 0.838611i \(-0.316632\pi\)
−0.529827 + 0.848106i \(0.677743\pi\)
\(30\) −0.0205702 + 0.0118762i −0.00375559 + 0.00216829i
\(31\) 4.32199 − 3.31638i 0.776252 − 0.595639i −0.142811 − 0.989750i \(-0.545614\pi\)
0.919063 + 0.394111i \(0.128947\pi\)
\(32\) 0.135172 − 0.0118261i 0.0238953 − 0.00209057i
\(33\) −1.90687 − 0.694046i −0.331944 − 0.120818i
\(34\) 0.00685619 − 0.0461187i 0.00117583 − 0.00790929i
\(35\) 2.16624 − 1.81769i 0.366162 − 0.307246i
\(36\) 3.70738 + 2.59593i 0.617896 + 0.432655i
\(37\) 9.66122 + 4.00181i 1.58830 + 0.657894i 0.989700 − 0.143158i \(-0.0457257\pi\)
0.598596 + 0.801051i \(0.295726\pi\)
\(38\) 0.0258998 + 0.0419392i 0.00420151 + 0.00680344i
\(39\) 1.42478 − 3.43971i 0.228147 − 0.550795i
\(40\) 0.0594649 + 0.0933412i 0.00940223 + 0.0147585i
\(41\) −1.73196 − 0.901602i −0.270487 − 0.140807i 0.321106 − 0.947043i \(-0.395945\pi\)
−0.591593 + 0.806237i \(0.701501\pi\)
\(42\) −0.0101681 − 0.00474146i −0.00156897 − 0.000731624i
\(43\) 0.911027 + 1.95370i 0.138930 + 0.297937i 0.963446 − 0.267902i \(-0.0863303\pi\)
−0.824516 + 0.565839i \(0.808552\pi\)
\(44\) −1.42158 + 4.50866i −0.214311 + 0.679707i
\(45\) −0.722763 + 5.48993i −0.107743 + 0.818391i
\(46\) −0.0518277 + 0.00682325i −0.00764158 + 0.00100603i
\(47\) 12.5155 + 2.20683i 1.82558 + 0.321899i 0.977974 − 0.208727i \(-0.0669321\pi\)
0.847606 + 0.530626i \(0.178043\pi\)
\(48\) −1.84461 + 2.89545i −0.266246 + 0.417922i
\(49\) −5.47129 − 1.46603i −0.781613 − 0.209433i
\(50\) −0.00557974 + 0.00966439i −0.000789094 + 0.00136675i
\(51\) 2.46066 + 2.54417i 0.344562 + 0.356255i
\(52\) −8.15051 − 2.96654i −1.13027 − 0.411386i
\(53\) −3.62072 + 7.76466i −0.497344 + 1.06656i 0.484136 + 0.874993i \(0.339134\pi\)
−0.981480 + 0.191565i \(0.938644\pi\)
\(54\) 0.0487270 − 0.0153636i 0.00663090 − 0.00209071i
\(55\) −5.69608 + 1.00437i −0.768060 + 0.135430i
\(56\) −0.0200051 + 0.0482965i −0.00267329 + 0.00645389i
\(57\) −3.72268 − 0.378291i −0.493082 − 0.0501058i
\(58\) −0.000411827 0 0.000994239i −5.40756e−5 0 0.000130550i
\(59\) −7.24544 + 10.3476i −0.943275 + 1.34714i −0.00546648 + 0.999985i \(0.501740\pi\)
−0.937809 + 0.347152i \(0.887149\pi\)
\(60\) −4.18461 − 0.366106i −0.540231 − 0.0472641i
\(61\) −2.73901 − 2.98911i −0.350694 − 0.382716i 0.536664 − 0.843796i \(-0.319684\pi\)
−0.887358 + 0.461080i \(0.847462\pi\)
\(62\) −0.0615464 + 0.00268718i −0.00781640 + 0.000341272i
\(63\) −2.31997 + 1.20770i −0.292289 + 0.152156i
\(64\) 6.92555 + 3.99847i 0.865693 + 0.499808i
\(65\) −1.38514 − 10.5212i −0.171805 − 1.30499i
\(66\) 0.0131622 + 0.0187975i 0.00162015 + 0.00231381i
\(67\) 1.02420 − 5.80851i 0.125126 − 0.709623i −0.856108 − 0.516798i \(-0.827124\pi\)
0.981233 − 0.192825i \(-0.0617650\pi\)
\(68\) 5.67135 − 5.98558i 0.687752 − 0.725858i
\(69\) 1.98415 − 3.43665i 0.238863 − 0.413724i
\(70\) −0.0318564 + 0.00278707i −0.00380757 + 0.000333119i
\(71\) −2.72624 − 2.49814i −0.323545 − 0.296475i 0.497354 − 0.867548i \(-0.334305\pi\)
−0.820899 + 0.571073i \(0.806527\pi\)
\(72\) −0.0350104 − 0.0961903i −0.00412602 − 0.0113361i
\(73\) 3.53091 + 11.1986i 0.413262 + 1.31070i 0.899518 + 0.436883i \(0.143918\pi\)
−0.486256 + 0.873816i \(0.661638\pi\)
\(74\) −0.0635379 − 0.0997345i −0.00738613 − 0.0115939i
\(75\) −0.324186 − 0.782654i −0.0374338 − 0.0903731i
\(76\) −0.502157 + 8.70277i −0.0576013 + 0.998276i
\(77\) −1.93181 − 1.93181i −0.220151 − 0.220151i
\(78\) −0.0355088 + 0.0226216i −0.00402058 + 0.00256139i
\(79\) 0.355563 − 0.683030i 0.0400039 − 0.0768469i −0.866176 − 0.499739i \(-0.833429\pi\)
0.906180 + 0.422892i \(0.138985\pi\)
\(80\) −0.426830 + 9.77601i −0.0477210 + 1.09299i
\(81\) 0.995542 − 2.73523i 0.110616 − 0.303914i
\(82\) 0.0101956 + 0.0195856i 0.00112592 + 0.00216287i
\(83\) 1.62433 + 6.06206i 0.178293 + 0.665398i 0.995967 + 0.0897171i \(0.0285963\pi\)
−0.817674 + 0.575681i \(0.804737\pi\)
\(84\) −0.992057 − 1.71829i −0.108242 − 0.187481i
\(85\) 9.69975 + 2.77328i 1.05209 + 0.300804i
\(86\) 0.00423304 − 0.0240068i 0.000456461 − 0.00258872i
\(87\) 0.0789096 + 0.0211438i 0.00846000 + 0.00226685i
\(88\) 0.0848279 − 0.0650907i 0.00904268 − 0.00693869i
\(89\) −8.69591 + 10.3634i −0.921765 + 1.09852i 0.0731019 + 0.997324i \(0.476710\pi\)
−0.994867 + 0.101192i \(0.967734\pi\)
\(90\) 0.0423040 − 0.0461667i 0.00445923 − 0.00486640i
\(91\) 3.69558 − 3.38637i 0.387402 − 0.354988i
\(92\) −8.20021 − 4.26876i −0.854931 − 0.445049i
\(93\) 2.68237 − 3.83082i 0.278149 − 0.397237i
\(94\) −0.101621 − 0.101621i −0.0104814 − 0.0104814i
\(95\) −9.72827 + 4.37158i −0.998099 + 0.448514i
\(96\) 0.107614 − 0.0445752i 0.0109833 − 0.00454944i
\(97\) −13.4200 − 2.97515i −1.36260 − 0.302080i −0.526272 − 0.850316i \(-0.676411\pi\)
−0.836324 + 0.548236i \(0.815300\pi\)
\(98\) 0.0411731 + 0.0490682i 0.00415911 + 0.00495663i
\(99\) 5.34457 + 0.233349i 0.537149 + 0.0234524i
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 323.2.bb.a.9.15 ✓ 672
17.2 even 8 inner 323.2.bb.a.104.14 yes 672
19.17 even 9 inner 323.2.bb.a.264.14 yes 672
323.36 even 72 inner 323.2.bb.a.36.15 yes 672
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
323.2.bb.a.9.15 ✓ 672 1.1 even 1 trivial
323.2.bb.a.36.15 yes 672 323.36 even 72 inner
323.2.bb.a.104.14 yes 672 17.2 even 8 inner
323.2.bb.a.264.14 yes 672 19.17 even 9 inner