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Properties

Label 444.2.ba.a.95.7
Level $444$
Weight $2$
Character 444.95
Analytic conductor $3.545$
Analytic rank $0$
Dimension $432$
Inner twists $8$

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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(95,444)] 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("444.95"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(444, base_ring=CyclotomicField(18)) chi = DirichletCharacter(H, H._module([9, 9, 11])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 444 = 2^{2} \cdot 3 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 444.ba (of order \(18\), degree \(6\), 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: \(3.54535784974\)
Analytic rank: \(0\)
Dimension: \(432\)
Relative dimension: \(72\) over \(\Q(\zeta_{18})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label 95.7
Character \(\chi\) \(=\) 444.95
Dual form 444.2.ba.a.215.7

$q$-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+(-1.37367 - 0.336197i) q^{2} +(-0.989335 - 1.42169i) q^{3} +(1.77394 + 0.923648i) q^{4} +(0.00719629 - 0.0408122i) q^{5} +(0.881051 + 2.28555i) q^{6} +(4.22835 + 0.745572i) q^{7} +(-2.12629 - 1.86518i) q^{8} +(-1.04243 + 2.81307i) q^{9} +(-0.0236063 + 0.0536432i) q^{10} +(1.33843 + 2.31822i) q^{11} +(-0.441879 - 3.43580i) q^{12} +(-2.38404 + 2.84119i) q^{13} +(-5.55770 - 2.44573i) q^{14} +(-0.0651420 + 0.0301460i) q^{15} +(2.29375 + 3.27700i) q^{16} +(-4.20943 + 3.53213i) q^{17} +(2.37770 - 3.51376i) q^{18} +(-2.23303 + 0.812758i) q^{19} +(0.0504619 - 0.0657517i) q^{20} +(-3.12328 - 6.74905i) q^{21} +(-1.05918 - 3.63445i) q^{22} +(7.44087 + 4.29599i) q^{23} +(-0.548109 + 4.86822i) q^{24} +(4.69685 + 1.70951i) q^{25} +(4.23009 - 3.10135i) q^{26} +(5.03063 - 1.30104i) q^{27} +(6.81221 + 5.22811i) q^{28} +(-0.172926 - 0.299517i) q^{29} +(0.0996187 - 0.0195102i) q^{30} -2.85873 q^{31} +(-2.04914 - 5.27267i) q^{32} +(1.97165 - 4.19633i) q^{33} +(6.96987 - 3.43679i) q^{34} +(0.0608569 - 0.167203i) q^{35} +(-4.44750 + 4.02738i) q^{36} +(-2.39403 - 5.59184i) q^{37} +(3.34070 - 0.365723i) q^{38} +(6.39793 + 0.578491i) q^{39} +(-0.0914236 + 0.0733561i) q^{40} +(-2.79321 + 3.32882i) q^{41} +(2.02135 + 10.3210i) q^{42} +5.20987 q^{43} +(0.233072 + 5.34863i) q^{44} +(0.107306 + 0.0627876i) q^{45} +(-8.77701 - 8.40288i) q^{46} +(5.78942 - 10.0276i) q^{47} +(2.38960 - 6.50306i) q^{48} +(10.7452 + 3.91094i) q^{49} +(-5.87719 - 3.92737i) q^{50} +(9.18615 + 2.49006i) q^{51} +(-6.85342 + 2.83810i) q^{52} +(11.9663 - 2.10998i) q^{53} +(-7.34784 + 0.0959220i) q^{54} +(0.104243 - 0.0379415i) q^{55} +(-7.60006 - 9.47195i) q^{56} +(3.36471 + 2.37060i) q^{57} +(0.136847 + 0.469575i) q^{58} +(3.12840 - 0.551621i) q^{59} +(-0.143403 - 0.00669097i) q^{60} +(-8.87023 + 10.5711i) q^{61} +(3.92696 + 0.961096i) q^{62} +(-6.50511 + 11.1174i) q^{63} +(1.04219 + 7.93182i) q^{64} +(0.0987990 + 0.117744i) q^{65} +(-4.11919 + 5.10151i) q^{66} +(-7.56440 - 1.33381i) q^{67} +(-10.7297 + 2.37777i) q^{68} +(-1.25393 - 14.8288i) q^{69} +(-0.139810 + 0.209222i) q^{70} +(-5.21505 + 1.89812i) q^{71} +(7.46339 - 4.03706i) q^{72} +0.821438 q^{73} +(1.40865 + 8.48621i) q^{74} +(-2.21635 - 8.36877i) q^{75} +(-4.71198 - 0.620750i) q^{76} +(3.93093 + 10.8001i) q^{77} +(-8.59416 - 2.94562i) q^{78} +(-0.330198 + 1.87265i) q^{79} +(0.150248 - 0.0700308i) q^{80} +(-6.82667 - 5.86486i) q^{81} +(4.95609 - 3.63363i) q^{82} +(4.73159 - 3.97028i) q^{83} +(0.693219 - 14.8572i) q^{84} +(0.113862 + 0.197214i) q^{85} +(-7.15664 - 1.75154i) q^{86} +(-0.254740 + 0.542172i) q^{87} +(1.47803 - 7.42561i) q^{88} +(0.348270 + 1.97514i) q^{89} +(-0.126294 - 0.122325i) q^{90} +(-12.1989 + 10.2361i) q^{91} +(9.23171 + 14.4936i) q^{92} +(2.82824 + 4.06424i) q^{93} +(-11.3240 + 11.8282i) q^{94} +(0.0171009 + 0.0969838i) q^{95} +(-5.46884 + 8.12969i) q^{96} +(2.61017 + 1.50698i) q^{97} +(-13.4456 - 8.98486i) q^{98} +(-7.91652 + 1.34849i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 432 q - 12 q^{4} - 12 q^{9} - 6 q^{10} - 21 q^{12} - 48 q^{13} - 36 q^{16} + 15 q^{18} - 12 q^{21} + 24 q^{22} - 51 q^{24} - 24 q^{25} - 36 q^{28} - 48 q^{30} + 6 q^{33} - 12 q^{34} - 12 q^{36} - 12 q^{37}+ \cdots - 36 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(149\) \(223\) \(409\)
\(\chi(n)\) \(-1\) \(-1\) \(e\left(\frac{11}{18}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.37367 − 0.336197i −0.971332 − 0.237727i
\(3\) −0.989335 − 1.42169i −0.571193 − 0.820816i
\(4\) 1.77394 + 0.923648i 0.886972 + 0.461824i
\(5\) 0.00719629 − 0.0408122i 0.00321828 − 0.0182518i −0.983156 − 0.182767i \(-0.941495\pi\)
0.986375 + 0.164515i \(0.0526058\pi\)
\(6\) 0.881051 + 2.28555i 0.359688 + 0.933073i
\(7\) 4.22835 + 0.745572i 1.59817 + 0.281800i 0.900577 − 0.434697i \(-0.143145\pi\)
0.697590 + 0.716497i \(0.254256\pi\)
\(8\) −2.12629 − 1.86518i −0.751756 − 0.659441i
\(9\) −1.04243 + 2.81307i −0.347477 + 0.937688i
\(10\) −0.0236063 + 0.0536432i −0.00746496 + 0.0169635i
\(11\) 1.33843 + 2.31822i 0.403550 + 0.698970i 0.994152 − 0.107994i \(-0.0344427\pi\)
−0.590601 + 0.806964i \(0.701109\pi\)
\(12\) −0.441879 − 3.43580i −0.127560 − 0.991831i
\(13\) −2.38404 + 2.84119i −0.661215 + 0.788005i −0.987559 − 0.157246i \(-0.949738\pi\)
0.326345 + 0.945251i \(0.394183\pi\)
\(14\) −5.55770 − 2.44573i −1.48536 − 0.653649i
\(15\) −0.0651420 + 0.0301460i −0.0168196 + 0.00778367i
\(16\) 2.29375 + 3.27700i 0.573438 + 0.819249i
\(17\) −4.20943 + 3.53213i −1.02094 + 0.856668i −0.989745 − 0.142845i \(-0.954375\pi\)
−0.0311922 + 0.999513i \(0.509930\pi\)
\(18\) 2.37770 − 3.51376i 0.560430 − 0.828202i
\(19\) −2.23303 + 0.812758i −0.512293 + 0.186459i −0.585215 − 0.810878i \(-0.698990\pi\)
0.0729219 + 0.997338i \(0.476768\pi\)
\(20\) 0.0504619 − 0.0657517i 0.0112836 − 0.0147025i
\(21\) −3.12328 − 6.74905i −0.681556 − 1.47276i
\(22\) −1.05918 − 3.63445i −0.225817 − 0.774867i
\(23\) 7.44087 + 4.29599i 1.55153 + 0.895776i 0.998018 + 0.0629348i \(0.0200460\pi\)
0.553512 + 0.832841i \(0.313287\pi\)
\(24\) −0.548109 + 4.86822i −0.111882 + 0.993721i
\(25\) 4.69685 + 1.70951i 0.939370 + 0.341903i
\(26\) 4.23009 − 3.10135i 0.829589 − 0.608226i
\(27\) 5.03063 − 1.30104i 0.968146 − 0.250386i
\(28\) 6.81221 + 5.22811i 1.28739 + 0.988020i
\(29\) −0.172926 − 0.299517i −0.0321116 − 0.0556190i 0.849523 − 0.527552i \(-0.176890\pi\)
−0.881635 + 0.471933i \(0.843557\pi\)
\(30\) 0.0996187 − 0.0195102i 0.0181878 − 0.00356205i
\(31\) −2.85873 −0.513443 −0.256722 − 0.966485i \(-0.582642\pi\)
−0.256722 + 0.966485i \(0.582642\pi\)
\(32\) −2.04914 − 5.27267i −0.362241 − 0.932085i
\(33\) 1.97165 − 4.19633i 0.343220 − 0.730487i
\(34\) 6.96987 − 3.43679i 1.19532 − 0.589405i
\(35\) 0.0608569 − 0.167203i 0.0102867 − 0.0282625i
\(36\) −4.44750 + 4.02738i −0.741249 + 0.671230i
\(37\) −2.39403 − 5.59184i −0.393576 − 0.919292i
\(38\) 3.34070 − 0.365723i 0.541933 − 0.0593281i
\(39\) 6.39793 + 0.578491i 1.02449 + 0.0926327i
\(40\) −0.0914236 + 0.0733561i −0.0144553 + 0.0115986i
\(41\) −2.79321 + 3.32882i −0.436226 + 0.519874i −0.938708 − 0.344714i \(-0.887976\pi\)
0.502482 + 0.864588i \(0.332420\pi\)
\(42\) 2.02135 + 10.3210i 0.311901 + 1.59257i
\(43\) 5.20987 0.794497 0.397249 − 0.917711i \(-0.369965\pi\)
0.397249 + 0.917711i \(0.369965\pi\)
\(44\) 0.233072 + 5.34863i 0.0351369 + 0.806336i
\(45\) 0.107306 + 0.0627876i 0.0159962 + 0.00935982i
\(46\) −8.77701 − 8.40288i −1.29410 − 1.23894i
\(47\) 5.78942 − 10.0276i 0.844473 − 1.46267i −0.0416051 − 0.999134i \(-0.513247\pi\)
0.886078 − 0.463536i \(-0.153420\pi\)
\(48\) 2.38960 − 6.50306i 0.344909 − 0.938636i
\(49\) 10.7452 + 3.91094i 1.53503 + 0.558706i
\(50\) −5.87719 − 3.92737i −0.831160 − 0.555415i
\(51\) 9.18615 + 2.49006i 1.28632 + 0.348679i
\(52\) −6.85342 + 2.83810i −0.950398 + 0.393573i
\(53\) 11.9663 − 2.10998i 1.64370 − 0.289828i 0.726173 − 0.687512i \(-0.241297\pi\)
0.917522 + 0.397684i \(0.130186\pi\)
\(54\) −7.34784 + 0.0959220i −0.999915 + 0.0130533i
\(55\) 0.104243 − 0.0379415i 0.0140562 − 0.00511603i
\(56\) −7.60006 − 9.47195i −1.01560 − 1.26574i
\(57\) 3.36471 + 2.37060i 0.445667 + 0.313994i
\(58\) 0.136847 + 0.469575i 0.0179689 + 0.0616583i
\(59\) 3.12840 − 0.551621i 0.407283 − 0.0718149i 0.0337462 − 0.999430i \(-0.489256\pi\)
0.373536 + 0.927616i \(0.378145\pi\)
\(60\) −0.143403 − 0.00669097i −0.0185132 − 0.000863801i
\(61\) −8.87023 + 10.5711i −1.13572 + 1.35349i −0.208919 + 0.977933i \(0.566995\pi\)
−0.926797 + 0.375562i \(0.877450\pi\)
\(62\) 3.92696 + 0.961096i 0.498724 + 0.122059i
\(63\) −6.50511 + 11.1174i −0.819567 + 1.40066i
\(64\) 1.04219 + 7.93182i 0.130274 + 0.991478i
\(65\) 0.0987990 + 0.117744i 0.0122545 + 0.0146044i
\(66\) −4.11919 + 5.10151i −0.507038 + 0.627953i
\(67\) −7.56440 − 1.33381i −0.924139 − 0.162951i −0.308722 − 0.951152i \(-0.599901\pi\)
−0.615417 + 0.788202i \(0.711012\pi\)
\(68\) −10.7297 + 2.37777i −1.30117 + 0.288347i
\(69\) −1.25393 − 14.8288i −0.150955 − 1.78518i
\(70\) −0.139810 + 0.209222i −0.0167105 + 0.0250068i
\(71\) −5.21505 + 1.89812i −0.618913 + 0.225266i −0.632399 − 0.774643i \(-0.717929\pi\)
0.0134856 + 0.999909i \(0.495707\pi\)
\(72\) 7.46339 − 4.03706i 0.879569 − 0.475772i
\(73\) 0.821438 0.0961421 0.0480710 − 0.998844i \(-0.484693\pi\)
0.0480710 + 0.998844i \(0.484693\pi\)
\(74\) 1.40865 + 8.48621i 0.163752 + 0.986502i
\(75\) −2.21635 − 8.36877i −0.255922 − 0.966342i
\(76\) −4.71198 − 0.620750i −0.540501 − 0.0712049i
\(77\) 3.93093 + 10.8001i 0.447971 + 1.23079i
\(78\) −8.59416 − 2.94562i −0.973097 − 0.333526i
\(79\) −0.330198 + 1.87265i −0.0371502 + 0.210689i −0.997732 − 0.0673073i \(-0.978559\pi\)
0.960582 + 0.277996i \(0.0896703\pi\)
\(80\) 0.150248 − 0.0700308i 0.0167982 − 0.00782968i
\(81\) −6.82667 − 5.86486i −0.758519 − 0.651651i
\(82\) 4.95609 − 3.63363i 0.547309 − 0.401268i
\(83\) 4.73159 − 3.97028i 0.519360 − 0.435795i −0.345049 − 0.938585i \(-0.612138\pi\)
0.864409 + 0.502790i \(0.167693\pi\)
\(84\) 0.693219 − 14.8572i 0.0756364 − 1.62106i
\(85\) 0.113862 + 0.197214i 0.0123500 + 0.0213909i
\(86\) −7.15664 − 1.75154i −0.771720 − 0.188873i
\(87\) −0.254740 + 0.542172i −0.0273110 + 0.0581269i
\(88\) 1.47803 − 7.42561i 0.157558 − 0.791573i
\(89\) 0.348270 + 1.97514i 0.0369166 + 0.209364i 0.997687 − 0.0679822i \(-0.0216561\pi\)
−0.960770 + 0.277347i \(0.910545\pi\)
\(90\) −0.126294 − 0.122325i −0.0133125 − 0.0128942i
\(91\) −12.1989 + 10.2361i −1.27879 + 1.07303i
\(92\) 9.23171 + 14.4936i 0.962472 + 1.51106i
\(93\) 2.82824 + 4.06424i 0.293275 + 0.421442i
\(94\) −11.3240 + 11.8282i −1.16798 + 1.21998i
\(95\) 0.0171009 + 0.0969838i 0.00175451 + 0.00995033i
\(96\) −5.46884 + 8.12969i −0.558161 + 0.829733i
\(97\) 2.61017 + 1.50698i 0.265023 + 0.153011i 0.626624 − 0.779322i \(-0.284436\pi\)
−0.361601 + 0.932333i \(0.617770\pi\)
\(98\) −13.4456 − 8.98486i −1.35821 − 0.907608i
\(99\) −7.91652 + 1.34849i −0.795640 + 0.135528i
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.ba.a.95.7 ✓ 432
3.2 odd 2 inner 444.2.ba.a.95.66 yes 432
4.3 odd 2 inner 444.2.ba.a.95.9 yes 432
12.11 even 2 inner 444.2.ba.a.95.64 yes 432
37.30 even 18 inner 444.2.ba.a.215.64 yes 432
111.104 odd 18 inner 444.2.ba.a.215.9 yes 432
148.67 odd 18 inner 444.2.ba.a.215.66 yes 432
444.215 even 18 inner 444.2.ba.a.215.7 yes 432
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
444.2.ba.a.95.7 ✓ 432 1.1 even 1 trivial
444.2.ba.a.95.9 yes 432 4.3 odd 2 inner
444.2.ba.a.95.64 yes 432 12.11 even 2 inner
444.2.ba.a.95.66 yes 432 3.2 odd 2 inner
444.2.ba.a.215.7 yes 432 444.215 even 18 inner
444.2.ba.a.215.9 yes 432 111.104 odd 18 inner
444.2.ba.a.215.64 yes 432 37.30 even 18 inner
444.2.ba.a.215.66 yes 432 148.67 odd 18 inner