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Properties

Label 444.2.c.d.371.13
Level $444$
Weight $2$
Character 444.371
Analytic conductor $3.545$
Analytic rank $0$
Dimension $36$
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] = [444,2,Mod(371,444)] 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("444.371"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(444, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 444 = 2^{2} \cdot 3 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 444.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [36,0] 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: \(3.54535784974\)
Analytic rank: \(0\)
Dimension: \(36\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 371.13
Character \(\chi\) \(=\) 444.371
Dual form 444.2.c.d.371.14

$q$-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+(-0.572407 - 1.29319i) q^{2} +(0.253239 - 1.71344i) q^{3} +(-1.34470 + 1.48047i) q^{4} -2.40717i q^{5} +(-2.36076 + 0.653297i) q^{6} +1.45151i q^{7} +(2.68425 + 0.891530i) q^{8} +(-2.87174 - 0.867817i) q^{9} +(-3.11294 + 1.37788i) q^{10} -2.02272 q^{11} +(2.19616 + 2.67897i) q^{12} -5.84894 q^{13} +(1.87708 - 0.830851i) q^{14} +(-4.12454 - 0.609588i) q^{15} +(-0.383559 - 3.98157i) q^{16} -5.73177i q^{17} +(0.521548 + 4.21046i) q^{18} +2.89367i q^{19} +(3.56373 + 3.23692i) q^{20} +(2.48706 + 0.367577i) q^{21} +(1.15782 + 2.61577i) q^{22} -0.553734 q^{23} +(2.20734 - 4.37352i) q^{24} -0.794464 q^{25} +(3.34797 + 7.56381i) q^{26} +(-2.21419 + 4.70078i) q^{27} +(-2.14890 - 1.95184i) q^{28} -0.211504i q^{29} +(1.57260 + 5.68276i) q^{30} +0.429852i q^{31} +(-4.92939 + 2.77509i) q^{32} +(-0.512232 + 3.46581i) q^{33} +(-7.41229 + 3.28090i) q^{34} +3.49402 q^{35} +(5.14641 - 3.08456i) q^{36} -1.00000 q^{37} +(3.74208 - 1.65636i) q^{38} +(-1.48118 + 10.0218i) q^{39} +(2.14606 - 6.46143i) q^{40} -4.35431i q^{41} +(-0.948264 - 3.42666i) q^{42} -3.22001i q^{43} +(2.71996 - 2.99457i) q^{44} +(-2.08898 + 6.91277i) q^{45} +(0.316961 + 0.716085i) q^{46} +7.82119 q^{47} +(-6.91930 - 0.351081i) q^{48} +4.89313 q^{49} +(0.454756 + 1.02740i) q^{50} +(-9.82103 - 1.45151i) q^{51} +(7.86507 - 8.65916i) q^{52} -3.07990i q^{53} +(7.34644 + 0.172611i) q^{54} +4.86904i q^{55} +(-1.29406 + 3.89620i) q^{56} +(4.95812 + 0.732788i) q^{57} +(-0.273516 + 0.121066i) q^{58} -12.6941 q^{59} +(6.44874 - 5.28652i) q^{60} -4.11312 q^{61} +(0.555882 - 0.246050i) q^{62} +(1.25964 - 4.16835i) q^{63} +(6.41035 + 4.78617i) q^{64} +14.0794i q^{65} +(4.77517 - 1.32144i) q^{66} -8.96023i q^{67} +(8.48569 + 7.70752i) q^{68} +(-0.140227 + 0.948789i) q^{69} +(-2.00000 - 4.51844i) q^{70} -8.76396 q^{71} +(-6.93477 - 4.88968i) q^{72} +13.2975 q^{73} +(0.572407 + 1.29319i) q^{74} +(-0.201189 + 1.36126i) q^{75} +(-4.28398 - 3.89112i) q^{76} -2.93599i q^{77} +(13.8080 - 3.82110i) q^{78} -16.4110i q^{79} +(-9.58431 + 0.923292i) q^{80} +(7.49379 + 4.98429i) q^{81} +(-5.63096 + 2.49244i) q^{82} -9.86090 q^{83} +(-3.88854 + 3.18773i) q^{84} -13.7973 q^{85} +(-4.16410 + 1.84316i) q^{86} +(-0.362399 - 0.0535610i) q^{87} +(-5.42949 - 1.80332i) q^{88} -7.27833i q^{89} +(10.1353 - 1.25545i) q^{90} -8.48977i q^{91} +(0.744606 - 0.819784i) q^{92} +(0.736525 + 0.108855i) q^{93} +(-4.47690 - 10.1143i) q^{94} +6.96555 q^{95} +(3.50664 + 9.14896i) q^{96} +6.69388 q^{97} +(-2.80086 - 6.32777i) q^{98} +(5.80874 + 1.75535i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 36 q - 2 q^{4} - 3 q^{6} - 2 q^{9} - 2 q^{10} - 11 q^{12} + 34 q^{16} + 21 q^{18} + 4 q^{21} - 8 q^{22} - 21 q^{24} - 32 q^{25} + 22 q^{28} + 28 q^{30} - 8 q^{33} - 42 q^{34} - 31 q^{36} - 36 q^{37} + 42 q^{40}+ \cdots + 32 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/444\mathbb{Z}\right)^\times\).

\(n\) \(149\) \(223\) \(409\)
\(\chi(n)\) \(-1\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.572407 − 1.29319i −0.404753 − 0.914426i
\(3\) 0.253239 − 1.71344i 0.146207 − 0.989254i
\(4\) −1.34470 + 1.48047i −0.672350 + 0.740233i
\(5\) − 2.40717i − 1.07652i −0.842779 − 0.538259i \(-0.819082\pi\)
0.842779 − 0.538259i \(-0.180918\pi\)
\(6\) −2.36076 + 0.653297i −0.963778 + 0.266707i
\(7\) 1.45151i 0.548617i 0.961642 + 0.274309i \(0.0884490\pi\)
−0.961642 + 0.274309i \(0.911551\pi\)
\(8\) 2.68425 + 0.891530i 0.949024 + 0.315203i
\(9\) −2.87174 − 0.867817i −0.957247 − 0.289272i
\(10\) −3.11294 + 1.37788i −0.984397 + 0.435724i
\(11\) −2.02272 −0.609874 −0.304937 − 0.952372i \(-0.598635\pi\)
−0.304937 + 0.952372i \(0.598635\pi\)
\(12\) 2.19616 + 2.67897i 0.633976 + 0.773353i
\(13\) −5.84894 −1.62220 −0.811102 − 0.584905i \(-0.801132\pi\)
−0.811102 + 0.584905i \(0.801132\pi\)
\(14\) 1.87708 − 0.830851i 0.501670 − 0.222054i
\(15\) −4.12454 − 0.609588i −1.06495 − 0.157395i
\(16\) −0.383559 − 3.98157i −0.0958898 − 0.995392i
\(17\) − 5.73177i − 1.39016i −0.718933 − 0.695079i \(-0.755369\pi\)
0.718933 − 0.695079i \(-0.244631\pi\)
\(18\) 0.521548 + 4.21046i 0.122930 + 0.992415i
\(19\) 2.89367i 0.663853i 0.943305 + 0.331927i \(0.107699\pi\)
−0.943305 + 0.331927i \(0.892301\pi\)
\(20\) 3.56373 + 3.23692i 0.796875 + 0.723798i
\(21\) 2.48706 + 0.367577i 0.542722 + 0.0802119i
\(22\) 1.15782 + 2.61577i 0.246848 + 0.557685i
\(23\) −0.553734 −0.115461 −0.0577307 − 0.998332i \(-0.518386\pi\)
−0.0577307 + 0.998332i \(0.518386\pi\)
\(24\) 2.20734 − 4.37352i 0.450571 − 0.892741i
\(25\) −0.794464 −0.158893
\(26\) 3.34797 + 7.56381i 0.656592 + 1.48339i
\(27\) −2.21419 + 4.70078i −0.426120 + 0.904666i
\(28\) −2.14890 − 1.95184i −0.406105 − 0.368863i
\(29\) − 0.211504i − 0.0392753i −0.999807 − 0.0196377i \(-0.993749\pi\)
0.999807 − 0.0196377i \(-0.00625127\pi\)
\(30\) 1.57260 + 5.68276i 0.287116 + 1.03752i
\(31\) 0.429852i 0.0772037i 0.999255 + 0.0386019i \(0.0122904\pi\)
−0.999255 + 0.0386019i \(0.987710\pi\)
\(32\) −4.92939 + 2.77509i −0.871401 + 0.490572i
\(33\) −0.512232 + 3.46581i −0.0891681 + 0.603321i
\(34\) −7.41229 + 3.28090i −1.27120 + 0.562671i
\(35\) 3.49402 0.590597
\(36\) 5.14641 − 3.08456i 0.857734 − 0.514093i
\(37\) −1.00000 −0.164399
\(38\) 3.74208 − 1.65636i 0.607045 − 0.268696i
\(39\) −1.48118 + 10.0218i −0.237178 + 1.60477i
\(40\) 2.14606 − 6.46143i 0.339322 − 1.02164i
\(41\) − 4.35431i − 0.680029i −0.940420 − 0.340014i \(-0.889568\pi\)
0.940420 − 0.340014i \(-0.110432\pi\)
\(42\) −0.948264 − 3.42666i −0.146320 − 0.528745i
\(43\) − 3.22001i − 0.491047i −0.969391 − 0.245524i \(-0.921040\pi\)
0.969391 − 0.245524i \(-0.0789599\pi\)
\(44\) 2.71996 − 2.99457i 0.410049 − 0.451449i
\(45\) −2.08898 + 6.91277i −0.311407 + 1.03049i
\(46\) 0.316961 + 0.716085i 0.0467334 + 0.105581i
\(47\) 7.82119 1.14084 0.570419 − 0.821354i \(-0.306781\pi\)
0.570419 + 0.821354i \(0.306781\pi\)
\(48\) −6.91930 − 0.351081i −0.998715 − 0.0506742i
\(49\) 4.89313 0.699019
\(50\) 0.454756 + 1.02740i 0.0643123 + 0.145296i
\(51\) −9.82103 − 1.45151i −1.37522 − 0.203251i
\(52\) 7.86507 − 8.65916i 1.09069 − 1.20081i
\(53\) − 3.07990i − 0.423057i −0.977372 − 0.211528i \(-0.932156\pi\)
0.977372 − 0.211528i \(-0.0678441\pi\)
\(54\) 7.34644 + 0.172611i 0.999724 + 0.0234893i
\(55\) 4.86904i 0.656541i
\(56\) −1.29406 + 3.89620i −0.172926 + 0.520651i
\(57\) 4.95812 + 0.732788i 0.656719 + 0.0970602i
\(58\) −0.273516 + 0.121066i −0.0359144 + 0.0158968i
\(59\) −12.6941 −1.65263 −0.826317 − 0.563205i \(-0.809568\pi\)
−0.826317 + 0.563205i \(0.809568\pi\)
\(60\) 6.44874 − 5.28652i 0.832529 − 0.682487i
\(61\) −4.11312 −0.526631 −0.263315 − 0.964710i \(-0.584816\pi\)
−0.263315 + 0.964710i \(0.584816\pi\)
\(62\) 0.555882 − 0.246050i 0.0705971 − 0.0312484i
\(63\) 1.25964 − 4.16835i 0.158700 − 0.525162i
\(64\) 6.41035 + 4.78617i 0.801294 + 0.598271i
\(65\) 14.0794i 1.74633i
\(66\) 4.77517 − 1.32144i 0.587783 − 0.162658i
\(67\) − 8.96023i − 1.09467i −0.836915 − 0.547333i \(-0.815643\pi\)
0.836915 − 0.547333i \(-0.184357\pi\)
\(68\) 8.48569 + 7.70752i 1.02904 + 0.934674i
\(69\) −0.140227 + 0.948789i −0.0168813 + 0.114221i
\(70\) −2.00000 − 4.51844i −0.239046 − 0.540057i
\(71\) −8.76396 −1.04009 −0.520045 − 0.854139i \(-0.674085\pi\)
−0.520045 + 0.854139i \(0.674085\pi\)
\(72\) −6.93477 − 4.88968i −0.817271 − 0.576254i
\(73\) 13.2975 1.55635 0.778177 − 0.628045i \(-0.216145\pi\)
0.778177 + 0.628045i \(0.216145\pi\)
\(74\) 0.572407 + 1.29319i 0.0665409 + 0.150331i
\(75\) −0.201189 + 1.36126i −0.0232313 + 0.157185i
\(76\) −4.28398 − 3.89112i −0.491406 − 0.446342i
\(77\) − 2.93599i − 0.334588i
\(78\) 13.8080 − 3.82110i 1.56344 − 0.432654i
\(79\) − 16.4110i − 1.84638i −0.384344 − 0.923190i \(-0.625572\pi\)
0.384344 − 0.923190i \(-0.374428\pi\)
\(80\) −9.58431 + 0.923292i −1.07156 + 0.103227i
\(81\) 7.49379 + 4.98429i 0.832643 + 0.553810i
\(82\) −5.63096 + 2.49244i −0.621836 + 0.275243i
\(83\) −9.86090 −1.08237 −0.541187 − 0.840902i \(-0.682025\pi\)
−0.541187 + 0.840902i \(0.682025\pi\)
\(84\) −3.88854 + 3.18773i −0.424275 + 0.347810i
\(85\) −13.7973 −1.49653
\(86\) −4.16410 + 1.84316i −0.449026 + 0.198753i
\(87\) −0.362399 − 0.0535610i −0.0388533 − 0.00574234i
\(88\) −5.42949 − 1.80332i −0.578786 − 0.192235i
\(89\) − 7.27833i − 0.771501i −0.922603 − 0.385751i \(-0.873942\pi\)
0.922603 − 0.385751i \(-0.126058\pi\)
\(90\) 10.1353 − 1.25545i 1.06835 − 0.132337i
\(91\) − 8.48977i − 0.889969i
\(92\) 0.744606 − 0.819784i 0.0776306 − 0.0854684i
\(93\) 0.736525 + 0.108855i 0.0763741 + 0.0112877i
\(94\) −4.47690 − 10.1143i −0.461757 − 1.04321i
\(95\) 6.96555 0.714650
\(96\) 3.50664 + 9.14896i 0.357895 + 0.933762i
\(97\) 6.69388 0.679661 0.339830 − 0.940487i \(-0.389630\pi\)
0.339830 + 0.940487i \(0.389630\pi\)
\(98\) −2.80086 − 6.32777i −0.282930 − 0.639201i
\(99\) 5.80874 + 1.75535i 0.583800 + 0.176420i
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 444.2.c.d.371.13 ✓ 36
3.2 odd 2 inner 444.2.c.d.371.24 yes 36
4.3 odd 2 inner 444.2.c.d.371.23 yes 36
12.11 even 2 inner 444.2.c.d.371.14 yes 36
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
444.2.c.d.371.13 ✓ 36 1.1 even 1 trivial
444.2.c.d.371.14 yes 36 12.11 even 2 inner
444.2.c.d.371.23 yes 36 4.3 odd 2 inner
444.2.c.d.371.24 yes 36 3.2 odd 2 inner