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Properties

Label 465.2.ba.a.4.14
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 4.14
Character \(\chi\) \(=\) 465.4
Dual form 465.2.ba.a.349.14

$q$-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+(-0.598879 + 0.194588i) q^{2} +(0.951057 + 0.309017i) q^{3} +(-1.29724 + 0.942501i) q^{4} +(1.97230 + 1.05357i) q^{5} -0.629699 q^{6} +(0.877498 + 1.20777i) q^{7} +(1.33375 - 1.83575i) q^{8} +(0.809017 + 0.587785i) q^{9} +(-1.38618 - 0.247178i) q^{10} +(3.45668 - 2.51143i) q^{11} +(-1.52500 + 0.495502i) q^{12} +(0.0347432 + 0.0112887i) q^{13} +(-0.760533 - 0.552559i) q^{14} +(1.55020 + 1.61148i) q^{15} +(0.549465 - 1.69108i) q^{16} +(-3.09546 + 4.26053i) q^{17} +(-0.598879 - 0.194588i) q^{18} +(0.148849 + 0.458111i) q^{19} +(-3.55155 + 0.492157i) q^{20} +(0.461328 + 1.41982i) q^{21} +(-1.58144 + 2.17667i) q^{22} +(-2.26822 + 3.12193i) q^{23} +(1.83575 - 1.33375i) q^{24} +(2.77996 + 4.15594i) q^{25} -0.0230036 q^{26} +(0.587785 + 0.809017i) q^{27} +(-2.27665 - 0.739730i) q^{28} +(1.42670 + 4.39095i) q^{29} +(-1.24196 - 0.663435i) q^{30} +(-0.998888 - 5.47743i) q^{31} +5.65788i q^{32} +(4.06358 - 1.32034i) q^{33} +(1.02476 - 3.15388i) q^{34} +(0.458214 + 3.30660i) q^{35} -1.60348 q^{36} -3.18896i q^{37} +(-0.178285 - 0.245389i) q^{38} +(0.0295543 + 0.0214725i) q^{39} +(4.56465 - 2.21545i) q^{40} +(1.18708 + 3.65347i) q^{41} +(-0.552559 - 0.760533i) q^{42} +(-2.02973 + 0.659499i) q^{43} +(-2.11713 + 6.51586i) q^{44} +(0.976351 + 2.01165i) q^{45} +(0.750898 - 2.31103i) q^{46} +(1.25092 + 0.406447i) q^{47} +(1.04514 - 1.43852i) q^{48} +(1.47441 - 4.53776i) q^{49} +(-2.47355 - 1.94796i) q^{50} +(-4.26053 + 3.09546i) q^{51} +(-0.0557100 + 0.0181013i) q^{52} +(3.10812 - 4.27796i) q^{53} +(-0.509437 - 0.370128i) q^{54} +(9.46361 - 1.31142i) q^{55} +3.38752 q^{56} +0.481686i q^{57} +(-1.70885 - 2.35203i) q^{58} +(-1.81110 + 5.57400i) q^{59} +(-3.52981 - 0.629420i) q^{60} -0.984217 q^{61} +(1.66405 + 3.08595i) q^{62} +1.49289i q^{63} +(-0.00202542 - 0.00623360i) q^{64} +(0.0566306 + 0.0588694i) q^{65} +(-2.17667 + 1.58144i) q^{66} -1.10338i q^{67} -8.44442i q^{68} +(-3.12193 + 2.26822i) q^{69} +(-0.917838 - 1.89109i) q^{70} +(-9.79602 - 7.11722i) q^{71} +(2.15805 - 0.701193i) q^{72} +(-8.26367 - 11.3740i) q^{73} +(0.620533 + 1.90980i) q^{74} +(1.35964 + 4.81159i) q^{75} +(-0.624864 - 0.453990i) q^{76} +(6.06646 + 1.97111i) q^{77} +(-0.0218777 - 0.00710851i) q^{78} +(4.26168 + 3.09629i) q^{79} +(2.86539 - 2.75642i) q^{80} +(0.309017 + 0.951057i) q^{81} +(-1.42184 - 1.95699i) q^{82} +(16.6299 - 5.40339i) q^{83} +(-1.93664 - 1.40705i) q^{84} +(-10.5940 + 5.14177i) q^{85} +(1.08723 - 0.789921i) q^{86} +4.61691i q^{87} -9.69521i q^{88} +(-13.0465 + 9.47884i) q^{89} +(-0.976159 - 1.01475i) q^{90} +(0.0168528 + 0.0518677i) q^{91} -6.18770i q^{92} +(0.742620 - 5.51802i) q^{93} -0.828238 q^{94} +(-0.189078 + 1.06036i) q^{95} +(-1.74838 + 5.38097i) q^{96} +(-4.70172 - 6.47137i) q^{97} +3.00447i q^{98} +4.27270 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.598879 + 0.194588i −0.423472 + 0.137594i −0.512997 − 0.858390i \(-0.671465\pi\)
0.0895257 + 0.995985i \(0.471465\pi\)
\(3\) 0.951057 + 0.309017i 0.549093 + 0.178411i
\(4\) −1.29724 + 0.942501i −0.648621 + 0.471251i
\(5\) 1.97230 + 1.05357i 0.882041 + 0.471173i
\(6\) −0.629699 −0.257073
\(7\) 0.877498 + 1.20777i 0.331663 + 0.456495i 0.941983 − 0.335660i \(-0.108959\pi\)
−0.610320 + 0.792155i \(0.708959\pi\)
\(8\) 1.33375 − 1.83575i 0.471551 − 0.649034i
\(9\) 0.809017 + 0.587785i 0.269672 + 0.195928i
\(10\) −1.38618 − 0.247178i −0.438350 − 0.0781646i
\(11\) 3.45668 − 2.51143i 1.04223 − 0.757224i 0.0715101 − 0.997440i \(-0.477218\pi\)
0.970720 + 0.240216i \(0.0772182\pi\)
\(12\) −1.52500 + 0.495502i −0.440229 + 0.143039i
\(13\) 0.0347432 + 0.0112887i 0.00963603 + 0.00313094i 0.313831 − 0.949479i \(-0.398387\pi\)
−0.304195 + 0.952610i \(0.598387\pi\)
\(14\) −0.760533 − 0.552559i −0.203261 − 0.147678i
\(15\) 1.55020 + 1.61148i 0.400260 + 0.416083i
\(16\) 0.549465 − 1.69108i 0.137366 − 0.422769i
\(17\) −3.09546 + 4.26053i −0.750759 + 1.03333i 0.247168 + 0.968973i \(0.420500\pi\)
−0.997927 + 0.0643584i \(0.979500\pi\)
\(18\) −0.598879 − 0.194588i −0.141157 − 0.0458648i
\(19\) 0.148849 + 0.458111i 0.0341484 + 0.105098i 0.966678 − 0.255996i \(-0.0824034\pi\)
−0.932529 + 0.361094i \(0.882403\pi\)
\(20\) −3.55155 + 0.492157i −0.794151 + 0.110050i
\(21\) 0.461328 + 1.41982i 0.100670 + 0.309830i
\(22\) −1.58144 + 2.17667i −0.337165 + 0.464068i
\(23\) −2.26822 + 3.12193i −0.472956 + 0.650968i −0.977132 − 0.212633i \(-0.931796\pi\)
0.504176 + 0.863601i \(0.331796\pi\)
\(24\) 1.83575 − 1.33375i 0.374720 − 0.272250i
\(25\) 2.77996 + 4.15594i 0.555992 + 0.831188i
\(26\) −0.0230036 −0.00451138
\(27\) 0.587785 + 0.809017i 0.113119 + 0.155695i
\(28\) −2.27665 − 0.739730i −0.430247 − 0.139796i
\(29\) 1.42670 + 4.39095i 0.264932 + 0.815378i 0.991709 + 0.128504i \(0.0410176\pi\)
−0.726777 + 0.686874i \(0.758982\pi\)
\(30\) −1.24196 − 0.663435i −0.226749 − 0.121126i
\(31\) −0.998888 − 5.47743i −0.179406 − 0.983775i
\(32\) 5.65788i 1.00018i
\(33\) 4.06358 − 1.32034i 0.707378 − 0.229841i
\(34\) 1.02476 − 3.15388i 0.175745 − 0.540886i
\(35\) 0.458214 + 3.30660i 0.0774522 + 0.558918i
\(36\) −1.60348 −0.267247
\(37\) − 3.18896i − 0.524262i −0.965032 − 0.262131i \(-0.915575\pi\)
0.965032 − 0.262131i \(-0.0844253\pi\)
\(38\) −0.178285 − 0.245389i −0.0289217 − 0.0398073i
\(39\) 0.0295543 + 0.0214725i 0.00473248 + 0.00343835i
\(40\) 4.56465 − 2.21545i 0.721735 − 0.350293i
\(41\) 1.18708 + 3.65347i 0.185391 + 0.570576i 0.999955 − 0.00949857i \(-0.00302354\pi\)
−0.814564 + 0.580074i \(0.803024\pi\)
\(42\) −0.552559 − 0.760533i −0.0852617 − 0.117353i
\(43\) −2.02973 + 0.659499i −0.309531 + 0.100573i −0.459663 − 0.888093i \(-0.652030\pi\)
0.150132 + 0.988666i \(0.452030\pi\)
\(44\) −2.11713 + 6.51586i −0.319170 + 0.982303i
\(45\) 0.976351 + 2.01165i 0.145546 + 0.299879i
\(46\) 0.750898 − 2.31103i 0.110714 − 0.340742i
\(47\) 1.25092 + 0.406447i 0.182465 + 0.0592864i 0.398824 − 0.917027i \(-0.369418\pi\)
−0.216359 + 0.976314i \(0.569418\pi\)
\(48\) 1.04514 − 1.43852i 0.150853 − 0.207632i
\(49\) 1.47441 − 4.53776i 0.210630 − 0.648252i
\(50\) −2.47355 − 1.94796i −0.349813 − 0.275483i
\(51\) −4.26053 + 3.09546i −0.596594 + 0.433451i
\(52\) −0.0557100 + 0.0181013i −0.00772559 + 0.00251019i
\(53\) 3.10812 − 4.27796i 0.426933 − 0.587623i −0.540313 − 0.841464i \(-0.681694\pi\)
0.967246 + 0.253841i \(0.0816942\pi\)
\(54\) −0.509437 − 0.370128i −0.0693256 − 0.0503680i
\(55\) 9.46361 − 1.31142i 1.27607 − 0.176832i
\(56\) 3.38752 0.452677
\(57\) 0.481686i 0.0638009i
\(58\) −1.70885 − 2.35203i −0.224383 − 0.308836i
\(59\) −1.81110 + 5.57400i −0.235785 + 0.725673i 0.761231 + 0.648481i \(0.224595\pi\)
−0.997016 + 0.0771917i \(0.975405\pi\)
\(60\) −3.52981 − 0.629420i −0.455697 − 0.0812578i
\(61\) −0.984217 −0.126016 −0.0630081 − 0.998013i \(-0.520069\pi\)
−0.0630081 + 0.998013i \(0.520069\pi\)
\(62\) 1.66405 + 3.08595i 0.211335 + 0.391916i
\(63\) 1.49289i 0.188086i
\(64\) −0.00202542 − 0.00623360i −0.000253177 − 0.000779200i
\(65\) 0.0566306 + 0.0588694i 0.00702416 + 0.00730185i
\(66\) −2.17667 + 1.58144i −0.267930 + 0.194662i
\(67\) − 1.10338i − 0.134799i −0.997726 − 0.0673995i \(-0.978530\pi\)
0.997726 − 0.0673995i \(-0.0214702\pi\)
\(68\) − 8.44442i − 1.02404i
\(69\) −3.12193 + 2.26822i −0.375836 + 0.273061i
\(70\) −0.917838 − 1.89109i −0.109703 − 0.226029i
\(71\) −9.79602 − 7.11722i −1.16257 − 0.844659i −0.172472 − 0.985014i \(-0.555175\pi\)
−0.990101 + 0.140355i \(0.955175\pi\)
\(72\) 2.15805 − 0.701193i 0.254329 − 0.0826363i
\(73\) −8.26367 − 11.3740i −0.967190 − 1.33122i −0.943454 − 0.331504i \(-0.892444\pi\)
−0.0237357 − 0.999718i \(-0.507556\pi\)
\(74\) 0.620533 + 1.90980i 0.0721355 + 0.222010i
\(75\) 1.35964 + 4.81159i 0.156998 + 0.555594i
\(76\) −0.624864 − 0.453990i −0.0716768 − 0.0520762i
\(77\) 6.06646 + 1.97111i 0.691338 + 0.224629i
\(78\) −0.0218777 − 0.00710851i −0.00247717 − 0.000804880i
\(79\) 4.26168 + 3.09629i 0.479477 + 0.348360i 0.801123 − 0.598500i \(-0.204236\pi\)
−0.321646 + 0.946860i \(0.604236\pi\)
\(80\) 2.86539 − 2.75642i 0.320360 − 0.308177i
\(81\) 0.309017 + 0.951057i 0.0343352 + 0.105673i
\(82\) −1.42184 − 1.95699i −0.157016 − 0.216114i
\(83\) 16.6299 − 5.40339i 1.82537 − 0.593099i 0.825795 − 0.563970i \(-0.190727\pi\)
0.999576 − 0.0291284i \(-0.00927318\pi\)
\(84\) −1.93664 − 1.40705i −0.211304 − 0.153522i
\(85\) −10.5940 + 5.14177i −1.14908 + 0.557703i
\(86\) 1.08723 − 0.789921i 0.117239 − 0.0851794i
\(87\) 4.61691i 0.494985i
\(88\) − 9.69521i − 1.03351i
\(89\) −13.0465 + 9.47884i −1.38293 + 1.00475i −0.386326 + 0.922362i \(0.626256\pi\)
−0.996600 + 0.0823925i \(0.973744\pi\)
\(90\) −0.976159 − 1.01475i −0.102896 − 0.106964i
\(91\) 0.0168528 + 0.0518677i 0.00176666 + 0.00543721i
\(92\) − 6.18770i − 0.645112i
\(93\) 0.742620 − 5.51802i 0.0770061 − 0.572192i
\(94\) −0.828238 −0.0854262
\(95\) −0.189078 + 1.06036i −0.0193990 + 0.108790i
\(96\) −1.74838 + 5.38097i −0.178444 + 0.549193i
\(97\) −4.70172 − 6.47137i −0.477388 − 0.657068i 0.500613 − 0.865671i \(-0.333108\pi\)
−0.978000 + 0.208604i \(0.933108\pi\)
\(98\) 3.00447i 0.303498i
\(99\) 4.27270 0.429422
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.4.14 ✓ 128
5.4 even 2 inner 465.2.ba.a.4.19 yes 128
31.8 even 5 inner 465.2.ba.a.349.19 yes 128
155.39 even 10 inner 465.2.ba.a.349.14 yes 128
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
465.2.ba.a.4.14 ✓ 128 1.1 even 1 trivial
465.2.ba.a.4.19 yes 128 5.4 even 2 inner
465.2.ba.a.349.14 yes 128 155.39 even 10 inner
465.2.ba.a.349.19 yes 128 31.8 even 5 inner