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Properties

Label 465.2.ba.a.4.13
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 4.13
Character \(\chi\) \(=\) 465.4
Dual form 465.2.ba.a.349.13

$q$-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} +(1.10611 + 1.71920i) 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} +(-0.126507 + 2.23249i) 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} +(2.01606 + 1.64680i) 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} +(1.52530 - 1.86732i) q^{35} -1.16417 q^{36} +9.73405i q^{37} +(-3.46549 - 4.76983i) q^{38} +(2.38209 + 1.73069i) q^{39} +(-6.02897 - 2.34372i) 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} +(0.810191 - 2.08413i) 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} +(3.08746 - 3.37096i) 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} +(2.97709 + 2.43180i) q^{55} +3.11925 q^{56} -6.44891i q^{57} +(-2.87996 - 3.96393i) q^{58} +(2.92785 - 9.01100i) q^{59} +(-1.40850 - 2.18920i) q^{60} +2.52330 q^{61} +(4.25666 - 2.79134i) q^{62} +1.07828i q^{63} +(-1.74833 - 5.38081i) q^{64} +(-0.372489 + 6.57339i) q^{65} +(-1.27151 + 0.923809i) q^{66} +13.4556i q^{67} +4.73934i q^{68} +(4.50709 - 3.27459i) q^{69} +(-0.798688 + 2.05454i) 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} +(4.89939 - 0.997991i) 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} +(0.706288 + 0.0400227i) 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} +(-8.48450 - 3.29829i) q^{85} +(8.88366 - 6.45436i) q^{86} -5.35931i q^{87} +4.97306i q^{88} +(-7.78574 + 5.65667i) q^{89} +(-0.115657 + 2.04102i) q^{90} +(-0.981101 - 3.01952i) q^{91} -6.48567i q^{92} +(5.56138 + 0.266632i) q^{93} +2.06911 q^{94} +(12.1270 - 7.80239i) 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 −0.599841 − 0.800120i \(-0.704769\pi\)
−0.0149827 + 0.999888i \(0.504769\pi\)
\(3\) −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 0.911818 − 0.410594i \(-0.134679\pi\)
−0.672266 + 0.740310i \(0.734679\pi\)
\(8\) 1.70035 − 2.34033i 0.601163 − 0.827430i
\(9\) 0.809017 + 0.587785i 0.269672 + 0.195928i
\(10\) 1.10611 + 1.71920i 0.349784 + 0.543660i
\(11\) −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 −0.106251 − 0.994339i \(-0.533885\pi\)
−0.670417 + 0.741985i \(0.733885\pi\)
\(14\) −0.797531 − 0.579440i −0.213149 − 0.154862i
\(15\) −0.126507 + 2.23249i −0.0326639 + 0.576426i
\(16\) −0.0977632 + 0.300884i −0.0244408 + 0.0752211i
\(17\) 2.39288 − 3.29351i 0.580358 − 0.798794i −0.413377 − 0.910560i \(-0.635651\pi\)
0.993735 + 0.111766i \(0.0356507\pi\)
\(18\) −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 \(0.634948\pi\)
\(20\) 2.01606 + 1.64680i 0.450806 + 0.368235i
\(21\) −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 −0.999963 − 0.00857759i \(-0.997270\pi\)
0.317163 + 0.948371i \(0.397270\pi\)
\(24\) −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
\(35\) 1.52530 − 1.86732i 0.257823 − 0.315635i
\(36\) −1.16417 −0.194028
\(37\) 9.73405i 1.60027i 0.599821 + 0.800134i \(0.295238\pi\)
−0.599821 + 0.800134i \(0.704762\pi\)
\(38\) −3.46549 − 4.76983i −0.562176 − 0.773769i
\(39\) 2.38209 + 1.73069i 0.381440 + 0.277132i
\(40\) −6.02897 − 2.34372i −0.953264 − 0.370575i
\(41\) 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 −0.995095 − 0.0989252i \(-0.968460\pi\)
−0.746902 + 0.664934i \(0.768460\pi\)
\(44\) 0.618448 − 1.90339i 0.0932346 − 0.286947i
\(45\) 0.810191 − 2.08413i 0.120776 − 0.310683i
\(46\) 1.57391 − 4.84400i 0.232060 − 0.714209i
\(47\) −2.15244 − 0.699371i −0.313966 − 0.102014i 0.147795 − 0.989018i \(-0.452782\pi\)
−0.461761 + 0.887004i \(0.652782\pi\)
\(48\) 0.185957 − 0.255947i 0.0268405 − 0.0369428i
\(49\) 1.80383 − 5.55162i 0.257690 − 0.793088i
\(50\) 3.08746 − 3.37096i 0.436632 − 0.476725i
\(51\) −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 −0.690144 − 0.723672i \(-0.742453\pi\)
0.901519 + 0.432739i \(0.142453\pi\)
\(54\) 0.739634 + 0.537375i 0.100651 + 0.0731275i
\(55\) 2.97709 + 2.43180i 0.401431 + 0.327905i
\(56\) 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 \(-0.111552\pi\)
\(60\) −1.40850 − 2.18920i −0.181837 − 0.282624i
\(61\) 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
\(65\) −0.372489 + 6.57339i −0.0462016 + 0.815328i
\(66\) −1.27151 + 0.923809i −0.156513 + 0.113713i
\(67\) 13.4556i 1.64386i 0.569586 + 0.821932i \(0.307104\pi\)
−0.569586 + 0.821932i \(0.692896\pi\)
\(68\) 4.73934i 0.574730i
\(69\) 4.50709 − 3.27459i 0.542590 − 0.394215i
\(70\) −0.798688 + 2.05454i −0.0954615 + 0.245564i
\(71\) 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 0.359418 − 0.933177i \(-0.382975\pi\)
−0.998570 + 0.0534592i \(0.982975\pi\)
\(74\) −2.75001 − 8.46367i −0.319683 − 0.983882i
\(75\) 4.89939 − 0.997991i 0.565733 − 0.115238i
\(76\) −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 \(0.520628\pi\)
\(80\) 0.706288 + 0.0400227i 0.0789654 + 0.00447467i
\(81\) 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 −0.340953 − 0.940080i \(-0.610750\pi\)
0.276729 + 0.960948i \(0.410750\pi\)
\(84\) 1.01556 + 0.737846i 0.110807 + 0.0805056i
\(85\) −8.48450 − 3.29829i −0.920273 − 0.357750i
\(86\) 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 \(0.470375\pi\)
\(90\) −0.115657 + 2.04102i −0.0121913 + 0.215143i
\(91\) −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
\(95\) 12.1270 − 7.80239i 1.24421 − 0.800508i
\(96\) 1.69847 − 5.22736i 0.173350 − 0.533515i
\(97\) 4.18696 + 5.76286i 0.425121 + 0.585129i 0.966825 − 0.255440i \(-0.0822204\pi\)
−0.541703 + 0.840570i \(0.682220\pi\)
\(98\) 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.4.13 ✓ 128
5.4 even 2 inner 465.2.ba.a.4.20 yes 128
31.8 even 5 inner 465.2.ba.a.349.20 yes 128
155.39 even 10 inner 465.2.ba.a.349.13 yes 128
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
465.2.ba.a.4.13 ✓ 128 1.1 even 1 trivial
465.2.ba.a.4.20 yes 128 5.4 even 2 inner
465.2.ba.a.349.13 yes 128 155.39 even 10 inner
465.2.ba.a.349.20 yes 128 31.8 even 5 inner