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Properties

Label 189.2.w.a.25.4
Level $189$
Weight $2$
Character 189.25
Analytic conductor $1.509$
Analytic rank $0$
Dimension $132$
Inner twists $2$

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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] = [189,2,Mod(25,189)] 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("189.25"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(189, base_ring=CyclotomicField(18)) chi = DirichletCharacter(H, H._module([10, 12])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 189 = 3^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 189.w (of order \(9\), 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: \(1.50917259820\)
Analytic rank: \(0\)
Dimension: \(132\)
Relative dimension: \(22\) over \(\Q(\zeta_{9})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

Embedding invariants

Embedding label 25.4
Character \(\chi\) \(=\) 189.25
Dual form 189.2.w.a.121.4

$q$-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+(-2.06527 + 0.751697i) q^{2} +(-0.131292 + 1.72707i) q^{3} +(2.16820 - 1.81934i) q^{4} +(-1.77012 - 0.644271i) q^{5} +(-1.02708 - 3.66555i) q^{6} +(-0.610341 - 2.57439i) q^{7} +(-0.912517 + 1.58053i) q^{8} +(-2.96552 - 0.453502i) q^{9} +4.14007 q^{10} +(-0.327911 + 0.119350i) q^{11} +(2.85745 + 3.98350i) q^{12} +(-0.602948 - 3.41949i) q^{13} +(3.19568 + 4.85802i) q^{14} +(1.34510 - 2.97253i) q^{15} +(-0.286465 + 1.62463i) q^{16} -0.361781 q^{17} +(6.46550 - 1.29257i) q^{18} +5.69812 q^{19} +(-5.01012 + 1.82354i) q^{20} +(4.52628 - 0.716103i) q^{21} +(0.587510 - 0.492979i) q^{22} +(-1.04537 - 5.92860i) q^{23} +(-2.60987 - 1.78349i) q^{24} +(-1.11198 - 0.933063i) q^{25} +(3.81566 + 6.60892i) q^{26} +(1.17258 - 5.06212i) q^{27} +(-6.00703 - 4.47138i) q^{28} +(1.01351 - 5.74790i) q^{29} +(-0.543560 + 7.15018i) q^{30} +(-4.32980 + 3.63313i) q^{31} +(-1.26342 - 7.16524i) q^{32} +(-0.163073 - 0.581994i) q^{33} +(0.747174 - 0.271949i) q^{34} +(-0.578228 + 4.95021i) q^{35} +(-7.25493 + 4.41201i) q^{36} +(-2.03435 + 3.52360i) q^{37} +(-11.7682 + 4.28326i) q^{38} +(5.98484 - 0.592379i) q^{39} +(2.63355 - 2.20981i) q^{40} +(1.73529 + 9.84132i) q^{41} +(-8.80969 + 4.88133i) q^{42} +(-7.48894 - 6.28397i) q^{43} +(-0.493839 + 0.855355i) q^{44} +(4.95716 + 2.71335i) q^{45} +(6.61549 + 11.4584i) q^{46} +(-1.77971 - 1.49336i) q^{47} +(-2.76823 - 0.708046i) q^{48} +(-6.25497 + 3.14251i) q^{49} +(2.99792 + 1.09115i) q^{50} +(0.0474990 - 0.624819i) q^{51} +(-7.52851 - 6.31717i) q^{52} +(2.00143 - 3.46658i) q^{53} +(1.38349 + 11.3361i) q^{54} +0.657336 q^{55} +(4.62584 + 1.38451i) q^{56} +(-0.748120 + 9.84104i) q^{57} +(2.22751 + 12.6328i) q^{58} +(-1.53680 - 8.71562i) q^{59} +(-2.49158 - 8.89224i) q^{60} +(-8.09969 - 6.79644i) q^{61} +(6.21119 - 10.7581i) q^{62} +(0.642492 + 7.91121i) q^{63} +(6.34571 + 10.9911i) q^{64} +(-1.13579 + 6.44136i) q^{65} +(0.774273 + 1.07939i) q^{66} +(-3.99124 - 1.45269i) q^{67} +(-0.784413 + 0.658201i) q^{68} +(10.3764 - 1.02705i) q^{69} +(-2.52686 - 10.6582i) q^{70} +(-5.31461 - 9.20518i) q^{71} +(3.42286 - 4.27326i) q^{72} +(1.59326 + 2.75961i) q^{73} +(1.55281 - 8.80641i) q^{74} +(1.75746 - 1.79796i) q^{75} +(12.3547 - 10.3668i) q^{76} +(0.507391 + 0.771327i) q^{77} +(-11.9150 + 5.72221i) q^{78} +(11.9365 - 4.34452i) q^{79} +(1.55378 - 2.69122i) q^{80} +(8.58867 + 2.68974i) q^{81} +(-10.9815 - 19.0206i) q^{82} +(-2.06563 + 11.7148i) q^{83} +(8.51105 - 9.78748i) q^{84} +(0.640395 + 0.233085i) q^{85} +(20.1903 + 7.34867i) q^{86} +(9.79395 + 2.50506i) q^{87} +(0.110589 - 0.627180i) q^{88} +11.2861 q^{89} +(-12.2775 - 1.87753i) q^{90} +(-8.43508 + 3.63927i) q^{91} +(-13.0527 - 10.9525i) q^{92} +(-5.70620 - 7.95486i) q^{93} +(4.79814 + 1.74638i) q^{94} +(-10.0864 - 3.67113i) q^{95} +(12.5407 - 1.24128i) q^{96} +(4.66375 + 3.91335i) q^{97} +(10.5560 - 11.1920i) q^{98} +(1.02655 - 0.205227i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 132 q - 3 q^{2} - 3 q^{3} - 3 q^{4} - 3 q^{5} - 18 q^{6} - 6 q^{7} - 6 q^{8} + 3 q^{9} - 6 q^{10} + 3 q^{11} - 3 q^{12} - 12 q^{13} + 15 q^{14} - 9 q^{16} - 54 q^{17} - 3 q^{18} - 6 q^{19} - 18 q^{20} - 21 q^{21}+ \cdots + 24 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/189\mathbb{Z}\right)^\times\).

\(n\) \(29\) \(136\)
\(\chi(n)\) \(e\left(\frac{5}{9}\right)\) \(e\left(\frac{2}{3}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.06527 + 0.751697i −1.46037 + 0.531530i −0.945466 − 0.325721i \(-0.894393\pi\)
−0.514900 + 0.857250i \(0.672171\pi\)
\(3\) −0.131292 + 1.72707i −0.0758017 + 0.997123i
\(4\) 2.16820 − 1.81934i 1.08410 − 0.909668i
\(5\) −1.77012 − 0.644271i −0.791622 − 0.288127i −0.0856118 − 0.996329i \(-0.527284\pi\)
−0.706010 + 0.708202i \(0.749507\pi\)
\(6\) −1.02708 − 3.66555i −0.419302 − 1.49646i
\(7\) −0.610341 − 2.57439i −0.230687 − 0.973028i
\(8\) −0.912517 + 1.58053i −0.322623 + 0.558800i
\(9\) −2.96552 − 0.453502i −0.988508 − 0.151167i
\(10\) 4.14007 1.30921
\(11\) −0.327911 + 0.119350i −0.0988689 + 0.0359853i −0.390981 − 0.920399i \(-0.627864\pi\)
0.292112 + 0.956384i \(0.405642\pi\)
\(12\) 2.85745 + 3.98350i 0.824875 + 1.14994i
\(13\) −0.602948 − 3.41949i −0.167228 − 0.948395i −0.946738 − 0.322005i \(-0.895643\pi\)
0.779510 − 0.626389i \(-0.215468\pi\)
\(14\) 3.19568 + 4.85802i 0.854081 + 1.29836i
\(15\) 1.34510 − 2.97253i 0.347304 − 0.767504i
\(16\) −0.286465 + 1.62463i −0.0716163 + 0.406156i
\(17\) −0.361781 −0.0877447 −0.0438723 − 0.999037i \(-0.513969\pi\)
−0.0438723 + 0.999037i \(0.513969\pi\)
\(18\) 6.46550 − 1.29257i 1.52393 − 0.304662i
\(19\) 5.69812 1.30724 0.653619 − 0.756824i \(-0.273250\pi\)
0.653619 + 0.756824i \(0.273250\pi\)
\(20\) −5.01012 + 1.82354i −1.12030 + 0.407755i
\(21\) 4.52628 − 0.716103i 0.987715 − 0.156266i
\(22\) 0.587510 − 0.492979i 0.125257 − 0.105103i
\(23\) −1.04537 − 5.92860i −0.217975 − 1.23620i −0.875668 − 0.482913i \(-0.839579\pi\)
0.657693 − 0.753286i \(-0.271532\pi\)
\(24\) −2.60987 − 1.78349i −0.532737 − 0.364053i
\(25\) −1.11198 − 0.933063i −0.222396 − 0.186613i
\(26\) 3.81566 + 6.60892i 0.748313 + 1.29612i
\(27\) 1.17258 − 5.06212i 0.225663 − 0.974205i
\(28\) −6.00703 − 4.47138i −1.13522 − 0.845011i
\(29\) 1.01351 − 5.74790i 0.188204 − 1.06736i −0.733565 − 0.679619i \(-0.762145\pi\)
0.921769 − 0.387739i \(-0.126744\pi\)
\(30\) −0.543560 + 7.15018i −0.0992400 + 1.30544i
\(31\) −4.32980 + 3.63313i −0.777655 + 0.652530i −0.942657 − 0.333763i \(-0.891681\pi\)
0.165002 + 0.986293i \(0.447237\pi\)
\(32\) −1.26342 − 7.16524i −0.223344 − 1.26665i
\(33\) −0.163073 − 0.581994i −0.0283874 − 0.101312i
\(34\) 0.747174 − 0.271949i 0.128139 − 0.0466389i
\(35\) −0.578228 + 4.95021i −0.0977384 + 0.836737i
\(36\) −7.25493 + 4.41201i −1.20915 + 0.735334i
\(37\) −2.03435 + 3.52360i −0.334446 + 0.579277i −0.983378 − 0.181569i \(-0.941882\pi\)
0.648933 + 0.760846i \(0.275216\pi\)
\(38\) −11.7682 + 4.28326i −1.90905 + 0.694836i
\(39\) 5.98484 − 0.592379i 0.958342 − 0.0948565i
\(40\) 2.63355 − 2.20981i 0.416401 − 0.349402i
\(41\) 1.73529 + 9.84132i 0.271007 + 1.53696i 0.751367 + 0.659884i \(0.229395\pi\)
−0.480361 + 0.877071i \(0.659494\pi\)
\(42\) −8.80969 + 4.88133i −1.35937 + 0.753206i
\(43\) −7.48894 − 6.28397i −1.14205 − 0.958296i −0.142548 − 0.989788i \(-0.545530\pi\)
−0.999504 + 0.0314920i \(0.989974\pi\)
\(44\) −0.493839 + 0.855355i −0.0744491 + 0.128950i
\(45\) 4.95716 + 2.71335i 0.738969 + 0.404483i
\(46\) 6.61549 + 11.4584i 0.975401 + 1.68944i
\(47\) −1.77971 − 1.49336i −0.259598 − 0.217828i 0.503694 − 0.863882i \(-0.331974\pi\)
−0.763292 + 0.646054i \(0.776418\pi\)
\(48\) −2.76823 − 0.708046i −0.399559 − 0.102198i
\(49\) −6.25497 + 3.14251i −0.893567 + 0.448930i
\(50\) 2.99792 + 1.09115i 0.423970 + 0.154313i
\(51\) 0.0474990 − 0.624819i 0.00665119 − 0.0874922i
\(52\) −7.52851 − 6.31717i −1.04402 − 0.876034i
\(53\) 2.00143 − 3.46658i 0.274918 − 0.476172i −0.695197 − 0.718820i \(-0.744683\pi\)
0.970114 + 0.242648i \(0.0780160\pi\)
\(54\) 1.38349 + 11.3361i 0.188269 + 1.54264i
\(55\) 0.657336 0.0886351
\(56\) 4.62584 + 1.38451i 0.618153 + 0.185014i
\(57\) −0.748120 + 9.84104i −0.0990909 + 1.30348i
\(58\) 2.22751 + 12.6328i 0.292486 + 1.65877i
\(59\) −1.53680 − 8.71562i −0.200074 − 1.13468i −0.905005 − 0.425401i \(-0.860133\pi\)
0.704931 − 0.709276i \(-0.250978\pi\)
\(60\) −2.49158 − 8.89224i −0.321661 − 1.14798i
\(61\) −8.09969 − 6.79644i −1.03706 − 0.870195i −0.0453845 − 0.998970i \(-0.514451\pi\)
−0.991674 + 0.128774i \(0.958896\pi\)
\(62\) 6.21119 − 10.7581i 0.788822 − 1.36628i
\(63\) 0.642492 + 7.91121i 0.0809463 + 0.996718i
\(64\) 6.34571 + 10.9911i 0.793214 + 1.37389i
\(65\) −1.13579 + 6.44136i −0.140877 + 0.798953i
\(66\) 0.774273 + 1.07939i 0.0953064 + 0.132864i
\(67\) −3.99124 − 1.45269i −0.487608 − 0.177475i 0.0865040 − 0.996252i \(-0.472430\pi\)
−0.574112 + 0.818777i \(0.694653\pi\)
\(68\) −0.784413 + 0.658201i −0.0951240 + 0.0798186i
\(69\) 10.3764 − 1.02705i 1.24917 − 0.123642i
\(70\) −2.52686 − 10.6582i −0.302017 − 1.27389i
\(71\) −5.31461 − 9.20518i −0.630728 − 1.09245i −0.987403 − 0.158225i \(-0.949423\pi\)
0.356675 − 0.934229i \(-0.383910\pi\)
\(72\) 3.42286 − 4.27326i 0.403388 − 0.503608i
\(73\) 1.59326 + 2.75961i 0.186477 + 0.322988i 0.944073 − 0.329736i \(-0.106960\pi\)
−0.757596 + 0.652724i \(0.773626\pi\)
\(74\) 1.55281 − 8.80641i 0.180510 − 1.02372i
\(75\) 1.75746 − 1.79796i 0.202934 − 0.207611i
\(76\) 12.3547 − 10.3668i 1.41718 − 1.18915i
\(77\) 0.507391 + 0.771327i 0.0578225 + 0.0879008i
\(78\) −11.9150 + 5.72221i −1.34911 + 0.647913i
\(79\) 11.9365 − 4.34452i 1.34296 − 0.488796i 0.432214 − 0.901771i \(-0.357732\pi\)
0.910743 + 0.412975i \(0.135510\pi\)
\(80\) 1.55378 − 2.69122i 0.173718 − 0.300888i
\(81\) 8.58867 + 2.68974i 0.954297 + 0.298860i
\(82\) −10.9815 − 19.0206i −1.21271 − 2.10047i
\(83\) −2.06563 + 11.7148i −0.226733 + 1.28587i 0.632612 + 0.774469i \(0.281983\pi\)
−0.859345 + 0.511397i \(0.829128\pi\)
\(84\) 8.51105 − 9.78748i 0.928632 − 1.06790i
\(85\) 0.640395 + 0.233085i 0.0694606 + 0.0252816i
\(86\) 20.1903 + 7.34867i 2.17718 + 0.792428i
\(87\) 9.79395 + 2.50506i 1.05002 + 0.268570i
\(88\) 0.110589 − 0.627180i 0.0117888 − 0.0668576i
\(89\) 11.2861 1.19633 0.598164 − 0.801374i \(-0.295897\pi\)
0.598164 + 0.801374i \(0.295897\pi\)
\(90\) −12.2775 − 1.87753i −1.29416 − 0.197909i
\(91\) −8.43508 + 3.63927i −0.884237 + 0.381500i
\(92\) −13.0527 − 10.9525i −1.36084 − 1.14188i
\(93\) −5.70620 − 7.95486i −0.591705 − 0.824881i
\(94\) 4.79814 + 1.74638i 0.494890 + 0.180125i
\(95\) −10.0864 − 3.67113i −1.03484 − 0.376650i
\(96\) 12.5407 − 1.24128i 1.27993 − 0.126688i
\(97\) 4.66375 + 3.91335i 0.473532 + 0.397341i 0.848081 − 0.529867i \(-0.177758\pi\)
−0.374549 + 0.927207i \(0.622202\pi\)
\(98\) 10.5560 − 11.1920i 1.06631 − 1.13056i
\(99\) 1.02655 − 0.205227i 0.103172 − 0.0206261i
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 189.2.w.a.25.4 yes 132
3.2 odd 2 567.2.w.a.235.19 132
7.2 even 3 189.2.u.a.79.19 yes 132
21.2 odd 6 567.2.u.a.478.4 132
27.13 even 9 189.2.u.a.67.19 ✓ 132
27.14 odd 18 567.2.u.a.172.4 132
189.121 even 9 inner 189.2.w.a.121.4 yes 132
189.149 odd 18 567.2.w.a.415.19 132
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
189.2.u.a.67.19 ✓ 132 27.13 even 9
189.2.u.a.79.19 yes 132 7.2 even 3
189.2.w.a.25.4 yes 132 1.1 even 1 trivial
189.2.w.a.121.4 yes 132 189.121 even 9 inner
567.2.u.a.172.4 132 27.14 odd 18
567.2.u.a.478.4 132 21.2 odd 6
567.2.w.a.235.19 132 3.2 odd 2
567.2.w.a.415.19 132 189.149 odd 18