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Properties

Label 465.2.ba.a.4.20
Level $465$
Weight $2$
Character 465.4
Analytic conductor $3.713$
Analytic rank $0$
Dimension $128$
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] = [465,2,Mod(4,465)] 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("465.4"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(465, base_ring=CyclotomicField(10)) chi = DirichletCharacter(H, H._module([0, 5, 6])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 465 = 3 \cdot 5 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 465.ba (of order \(10\), degree \(4\), 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: \(3.71304369399\)
Analytic rank: \(0\)
Dimension: \(128\)
Relative dimension: \(32\) over \(\Q(\zeta_{10})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

Embedding invariants

Embedding label 4.20
Character \(\chi\) \(=\) 465.4
Dual form 465.2.ba.a.349.20

$q$-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+(0.869491 - 0.282515i) q^{2} +(0.951057 + 0.309017i) q^{3} +(-0.941833 + 0.684282i) q^{4} +(-0.569561 + 2.16231i) q^{5} +0.914237 q^{6} +(-0.633796 - 0.872346i) q^{7} +(-1.70035 + 2.34033i) q^{8} +(0.809017 + 0.587785i) q^{9} +(0.115657 + 2.04102i) q^{10} +(-1.39079 + 1.01047i) q^{11} +(-1.10719 + 0.359748i) q^{12} +(2.80031 + 0.909877i) q^{13} +(-0.797531 - 0.579440i) q^{14} +(-1.20988 + 1.88048i) q^{15} +(-0.0977632 + 0.300884i) q^{16} +(-2.39288 + 3.29351i) q^{17} +(0.869491 + 0.282515i) q^{18} +(1.99282 + 6.13328i) q^{19} +(-0.943200 - 2.42628i) q^{20} +(-0.333206 - 1.02550i) q^{21} +(-0.923809 + 1.27151i) q^{22} +(3.27459 - 4.50709i) q^{23} +(-2.34033 + 1.70035i) q^{24} +(-4.35120 - 2.46314i) q^{25} +2.69190 q^{26} +(0.587785 + 0.809017i) q^{27} +(1.19386 + 0.387909i) q^{28} +(1.65612 + 5.09701i) q^{29} +(-0.520714 + 1.97687i) q^{30} +(-5.37158 + 1.46498i) q^{31} -5.49637i q^{32} +(-1.63497 + 0.531235i) q^{33} +(-1.15012 + 3.53970i) q^{34} +(2.24727 - 0.873612i) q^{35} -1.16417 q^{36} -9.73405i q^{37} +(3.46549 + 4.76983i) q^{38} +(2.38209 + 1.73069i) q^{39} +(-4.09207 - 5.00964i) q^{40} +(1.26884 + 3.90509i) q^{41} +(-0.579440 - 0.797531i) q^{42} +(11.4230 - 3.71157i) q^{43} +(0.618448 - 1.90339i) q^{44} +(-1.73176 + 1.41457i) q^{45} +(1.57391 - 4.84400i) q^{46} +(2.15244 + 0.699371i) q^{47} +(-0.185957 + 0.255947i) q^{48} +(1.80383 - 5.55162i) q^{49} +(-4.47920 - 0.912401i) q^{50} +(-3.29351 + 2.39288i) q^{51} +(-3.26004 + 1.05925i) q^{52} +(-1.53883 + 2.11802i) q^{53} +(0.739634 + 0.537375i) q^{54} +(-1.39281 - 3.58285i) q^{55} +3.11925 q^{56} +6.44891i q^{57} +(2.87996 + 3.96393i) q^{58} +(2.92785 - 9.01100i) q^{59} +(-0.147275 - 2.59899i) q^{60} +2.52330 q^{61} +(-4.25666 + 2.79134i) q^{62} -1.07828i q^{63} +(-1.74833 - 5.38081i) q^{64} +(-3.56239 + 5.53693i) q^{65} +(-1.27151 + 0.923809i) q^{66} -13.4556i q^{67} -4.73934i q^{68} +(4.50709 - 3.27459i) q^{69} +(1.70717 - 1.39449i) q^{70} +(10.9754 + 7.97409i) q^{71} +(-2.75122 + 0.893925i) q^{72} +(5.46092 + 7.51631i) q^{73} +(-2.75001 - 8.46367i) q^{74} +(-3.37708 - 3.68718i) q^{75} +(-6.07380 - 4.41287i) q^{76} +(1.76296 + 0.572820i) q^{77} +(2.56015 + 0.831844i) q^{78} +(7.68199 + 5.58129i) q^{79} +(-0.594924 - 0.382767i) q^{80} +(0.309017 + 0.951057i) q^{81} +(2.20649 + 3.03697i) q^{82} +(0.585109 - 0.190114i) q^{83} +(1.01556 + 0.737846i) q^{84} +(-5.75872 - 7.05001i) q^{85} +(8.88366 - 6.45436i) q^{86} +5.35931i q^{87} -4.97306i q^{88} +(-7.78574 + 5.65667i) q^{89} +(-1.10611 + 1.71920i) q^{90} +(-0.981101 - 3.01952i) q^{91} +6.48567i q^{92} +(-5.56138 - 0.266632i) q^{93} +2.06911 q^{94} +(-14.3971 + 0.815830i) q^{95} +(1.69847 - 5.22736i) q^{96} +(-4.18696 - 5.76286i) q^{97} -5.33669i q^{98} -1.71911 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 128 q + 36 q^{4} - 20 q^{6} + 32 q^{9} + 2 q^{10} + 4 q^{14} - 4 q^{15} - 40 q^{16} - 4 q^{19} + 36 q^{20} + 20 q^{21} - 20 q^{24} + 12 q^{25} + 24 q^{26} - 28 q^{29} - 8 q^{30} - 12 q^{31} - 44 q^{34}+ \cdots - 20 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/465\mathbb{Z}\right)^\times\).

\(n\) \(187\) \(311\) \(406\)
\(\chi(n)\) \(-1\) \(1\) \(e\left(\frac{3}{5}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.869491 − 0.282515i 0.614823 − 0.199768i 0.0149827 − 0.999888i \(-0.495231\pi\)
0.599841 + 0.800120i \(0.295231\pi\)
\(3\) 0.951057 + 0.309017i 0.549093 + 0.178411i
\(4\) −0.941833 + 0.684282i −0.470917 + 0.342141i
\(5\) −0.569561 + 2.16231i −0.254716 + 0.967016i
\(6\) 0.914237 0.373236
\(7\) −0.633796 − 0.872346i −0.239552 − 0.329716i 0.672266 − 0.740310i \(-0.265321\pi\)
−0.911818 + 0.410594i \(0.865321\pi\)
\(8\) −1.70035 + 2.34033i −0.601163 + 0.827430i
\(9\) 0.809017 + 0.587785i 0.269672 + 0.195928i
\(10\) 0.115657 + 2.04102i 0.0365740 + 0.645428i
\(11\) −1.39079 + 1.01047i −0.419340 + 0.304668i −0.777372 − 0.629041i \(-0.783448\pi\)
0.358032 + 0.933709i \(0.383448\pi\)
\(12\) −1.10719 + 0.359748i −0.319619 + 0.103850i
\(13\) 2.80031 + 0.909877i 0.776667 + 0.252354i 0.670417 − 0.741985i \(-0.266115\pi\)
0.106251 + 0.994339i \(0.466115\pi\)
\(14\) −0.797531 − 0.579440i −0.213149 − 0.154862i
\(15\) −1.20988 + 1.88048i −0.312389 + 0.485537i
\(16\) −0.0977632 + 0.300884i −0.0244408 + 0.0752211i
\(17\) −2.39288 + 3.29351i −0.580358 + 0.798794i −0.993735 − 0.111766i \(-0.964349\pi\)
0.413377 + 0.910560i \(0.364349\pi\)
\(18\) 0.869491 + 0.282515i 0.204941 + 0.0665894i
\(19\) 1.99282 + 6.13328i 0.457185 + 1.40707i 0.868551 + 0.495600i \(0.165052\pi\)
−0.411366 + 0.911470i \(0.634948\pi\)
\(20\) −0.943200 − 2.42628i −0.210906 − 0.542533i
\(21\) −0.333206 − 1.02550i −0.0727116 − 0.223783i
\(22\) −0.923809 + 1.27151i −0.196957 + 0.271088i
\(23\) 3.27459 − 4.50709i 0.682800 − 0.939793i −0.317163 − 0.948371i \(-0.602730\pi\)
0.999963 + 0.00857759i \(0.00273037\pi\)
\(24\) −2.34033 + 1.70035i −0.477717 + 0.347082i
\(25\) −4.35120 − 2.46314i −0.870240 − 0.492628i
\(26\) 2.69190 0.527926
\(27\) 0.587785 + 0.809017i 0.113119 + 0.155695i
\(28\) 1.19386 + 0.387909i 0.225618 + 0.0733079i
\(29\) 1.65612 + 5.09701i 0.307533 + 0.946491i 0.978720 + 0.205202i \(0.0657851\pi\)
−0.671186 + 0.741289i \(0.734215\pi\)
\(30\) −0.520714 + 1.97687i −0.0950690 + 0.360925i
\(31\) −5.37158 + 1.46498i −0.964764 + 0.263118i
\(32\) − 5.49637i − 0.971630i
\(33\) −1.63497 + 0.531235i −0.284613 + 0.0924762i
\(34\) −1.15012 + 3.53970i −0.197244 + 0.607054i
\(35\) 2.24727 − 0.873612i 0.379858 − 0.147667i
\(36\) −1.16417 −0.194028
\(37\) − 9.73405i − 1.60027i −0.599821 − 0.800134i \(-0.704762\pi\)
0.599821 − 0.800134i \(-0.295238\pi\)
\(38\) 3.46549 + 4.76983i 0.562176 + 0.773769i
\(39\) 2.38209 + 1.73069i 0.381440 + 0.277132i
\(40\) −4.09207 − 5.00964i −0.647013 − 0.792094i
\(41\) 1.26884 + 3.90509i 0.198159 + 0.609872i 0.999925 + 0.0122335i \(0.00389415\pi\)
−0.801766 + 0.597638i \(0.796106\pi\)
\(42\) −0.579440 − 0.797531i −0.0894096 − 0.123062i
\(43\) 11.4230 − 3.71157i 1.74200 − 0.566009i 0.746902 − 0.664934i \(-0.231540\pi\)
0.995095 + 0.0989252i \(0.0315405\pi\)
\(44\) 0.618448 − 1.90339i 0.0932346 − 0.286947i
\(45\) −1.73176 + 1.41457i −0.258156 + 0.210871i
\(46\) 1.57391 − 4.84400i 0.232060 − 0.714209i
\(47\) 2.15244 + 0.699371i 0.313966 + 0.102014i 0.461761 − 0.887004i \(-0.347218\pi\)
−0.147795 + 0.989018i \(0.547218\pi\)
\(48\) −0.185957 + 0.255947i −0.0268405 + 0.0369428i
\(49\) 1.80383 − 5.55162i 0.257690 − 0.793088i
\(50\) −4.47920 − 0.912401i −0.633455 − 0.129033i
\(51\) −3.29351 + 2.39288i −0.461184 + 0.335070i
\(52\) −3.26004 + 1.05925i −0.452086 + 0.146892i
\(53\) −1.53883 + 2.11802i −0.211375 + 0.290933i −0.901519 − 0.432739i \(-0.857547\pi\)
0.690144 + 0.723672i \(0.257547\pi\)
\(54\) 0.739634 + 0.537375i 0.100651 + 0.0731275i
\(55\) −1.39281 − 3.58285i −0.187807 − 0.483112i
\(56\) 3.11925 0.416827
\(57\) 6.44891i 0.854179i
\(58\) 2.87996 + 3.96393i 0.378158 + 0.520489i
\(59\) 2.92785 − 9.01100i 0.381174 − 1.17313i −0.558045 − 0.829811i \(-0.688448\pi\)
0.939218 − 0.343321i \(-0.111552\pi\)
\(60\) −0.147275 − 2.59899i −0.0190131 − 0.335529i
\(61\) 2.52330 0.323076 0.161538 − 0.986866i \(-0.448355\pi\)
0.161538 + 0.986866i \(0.448355\pi\)
\(62\) −4.25666 + 2.79134i −0.540597 + 0.354500i
\(63\) − 1.07828i − 0.135850i
\(64\) −1.74833 − 5.38081i −0.218542 − 0.672602i
\(65\) −3.56239 + 5.53693i −0.441860 + 0.686771i
\(66\) −1.27151 + 0.923809i −0.156513 + 0.113713i
\(67\) − 13.4556i − 1.64386i −0.569586 − 0.821932i \(-0.692896\pi\)
0.569586 − 0.821932i \(-0.307104\pi\)
\(68\) − 4.73934i − 0.574730i
\(69\) 4.50709 − 3.27459i 0.542590 − 0.394215i
\(70\) 1.70717 − 1.39449i 0.204046 − 0.166673i
\(71\) 10.9754 + 7.97409i 1.30254 + 0.946351i 0.999977 − 0.00677870i \(-0.00215774\pi\)
0.302563 + 0.953129i \(0.402158\pi\)
\(72\) −2.75122 + 0.893925i −0.324234 + 0.105350i
\(73\) 5.46092 + 7.51631i 0.639152 + 0.879718i 0.998570 − 0.0534592i \(-0.0170247\pi\)
−0.359418 + 0.933177i \(0.617025\pi\)
\(74\) −2.75001 − 8.46367i −0.319683 − 0.983882i
\(75\) −3.37708 − 3.68718i −0.389952 − 0.425759i
\(76\) −6.07380 − 4.41287i −0.696713 − 0.506191i
\(77\) 1.76296 + 0.572820i 0.200908 + 0.0652789i
\(78\) 2.56015 + 0.831844i 0.289880 + 0.0941877i
\(79\) 7.68199 + 5.58129i 0.864291 + 0.627944i 0.929049 − 0.369957i \(-0.120628\pi\)
−0.0647579 + 0.997901i \(0.520628\pi\)
\(80\) −0.594924 − 0.382767i −0.0665145 − 0.0427946i
\(81\) 0.309017 + 0.951057i 0.0343352 + 0.105673i
\(82\) 2.20649 + 3.03697i 0.243666 + 0.335378i
\(83\) 0.585109 − 0.190114i 0.0642241 − 0.0208677i −0.276729 − 0.960948i \(-0.589250\pi\)
0.340953 + 0.940080i \(0.389250\pi\)
\(84\) 1.01556 + 0.737846i 0.110807 + 0.0805056i
\(85\) −5.75872 − 7.05001i −0.624621 − 0.764681i
\(86\) 8.88366 − 6.45436i 0.957950 − 0.695991i
\(87\) 5.35931i 0.574579i
\(88\) − 4.97306i − 0.530130i
\(89\) −7.78574 + 5.65667i −0.825287 + 0.599606i −0.918222 − 0.396066i \(-0.870375\pi\)
0.0929354 + 0.995672i \(0.470375\pi\)
\(90\) −1.10611 + 1.71920i −0.116595 + 0.181220i
\(91\) −0.981101 − 3.01952i −0.102847 − 0.316531i
\(92\) 6.48567i 0.676178i
\(93\) −5.56138 − 0.266632i −0.576688 − 0.0276485i
\(94\) 2.06911 0.213413
\(95\) −14.3971 + 0.815830i −1.47711 + 0.0837024i
\(96\) 1.69847 − 5.22736i 0.173350 − 0.533515i
\(97\) −4.18696 − 5.76286i −0.425121 − 0.585129i 0.541703 − 0.840570i \(-0.317780\pi\)
−0.966825 + 0.255440i \(0.917780\pi\)
\(98\) − 5.33669i − 0.539087i
\(99\) −1.71911 −0.172777
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 465.2.ba.a.4.20 yes 128
5.4 even 2 inner 465.2.ba.a.4.13 ✓ 128
31.8 even 5 inner 465.2.ba.a.349.13 yes 128
155.39 even 10 inner 465.2.ba.a.349.20 yes 128
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
465.2.ba.a.4.13 ✓ 128 5.4 even 2 inner
465.2.ba.a.4.20 yes 128 1.1 even 1 trivial
465.2.ba.a.349.13 yes 128 31.8 even 5 inner
465.2.ba.a.349.20 yes 128 155.39 even 10 inner