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Properties

Label 465.2.ba.a.4.19
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 4.19
Character \(\chi\) \(=\) 465.4
Dual form 465.2.ba.a.349.19

$q$-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} +(0.976159 - 1.01475i) 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} +(-2.20134 + 0.392534i) 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} +(-1.56556 + 3.22564i) 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} +(-3.00317 - 1.45758i) q^{35} -1.60348 q^{36} +3.18896i q^{37} +(0.178285 + 0.245389i) q^{38} +(0.0295543 + 0.0214725i) q^{39} +(-0.696459 + 5.02585i) 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} +(2.21490 + 0.306931i) 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} +(0.856166 - 3.02985i) 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} +(4.17165 - 8.59517i) q^{55} +3.38752 q^{56} -0.481686i q^{57} +(1.70885 + 2.35203i) q^{58} +(-1.81110 + 5.57400i) q^{59} +(2.48571 - 2.58398i) q^{60} -0.984217 q^{61} +(-1.66405 - 3.08595i) q^{62} -1.49289i q^{63} +(-0.00202542 - 0.00623360i) q^{64} +(-0.0804176 + 0.0143397i) q^{65} +(-2.17667 + 1.58144i) q^{66} +1.10338i q^{67} +8.44442i q^{68} +(-3.12193 + 2.26822i) q^{69} +(-2.08216 - 0.288537i) 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} +(-3.92816 + 3.09348i) 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} +(-0.697966 - 3.91422i) 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} +(1.61639 - 11.6644i) q^{85} +(1.08723 - 0.789921i) q^{86} -4.61691i q^{87} +9.69521i q^{88} +(-13.0465 + 9.47884i) q^{89} +(1.38618 - 0.247178i) q^{90} +(0.0168528 + 0.0518677i) q^{91} +6.18770i q^{92} +(-0.742620 + 5.51802i) q^{93} -0.828238 q^{94} +(0.776230 + 0.746710i) 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 −0.0895257 − 0.995985i \(-0.528535\pi\)
0.512997 + 0.858390i \(0.328535\pi\)
\(3\) −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 0.610320 − 0.792155i \(-0.291041\pi\)
−0.941983 + 0.335660i \(0.891041\pi\)
\(8\) −1.33375 + 1.83575i −0.471551 + 0.649034i
\(9\) 0.809017 + 0.587785i 0.269672 + 0.195928i
\(10\) 0.976159 − 1.01475i 0.308689 − 0.320892i
\(11\) 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 0.304195 − 0.952610i \(-0.401613\pi\)
−0.313831 + 0.949479i \(0.601613\pi\)
\(14\) −0.760533 − 0.552559i −0.203261 − 0.147678i
\(15\) −2.20134 + 0.392534i −0.568385 + 0.101352i
\(16\) 0.549465 − 1.69108i 0.137366 − 0.422769i
\(17\) 3.09546 − 4.26053i 0.750759 − 1.03333i −0.247168 − 0.968973i \(-0.579500\pi\)
0.997927 − 0.0643584i \(-0.0205001\pi\)
\(18\) 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 \(0.882403\pi\)
\(20\) −1.56556 + 3.22564i −0.350070 + 0.721275i
\(21\) 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 −0.504176 − 0.863601i \(-0.668204\pi\)
0.977132 + 0.212633i \(0.0682039\pi\)
\(24\) 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
\(35\) −3.00317 − 1.45758i −0.507628 − 0.246376i
\(36\) −1.60348 −0.267247
\(37\) 3.18896i 0.524262i 0.965032 + 0.262131i \(0.0844253\pi\)
−0.965032 + 0.262131i \(0.915575\pi\)
\(38\) 0.178285 + 0.245389i 0.0289217 + 0.0398073i
\(39\) 0.0295543 + 0.0214725i 0.00473248 + 0.00343835i
\(40\) −0.696459 + 5.02585i −0.110120 + 0.794657i
\(41\) 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 −0.150132 − 0.988666i \(-0.547970\pi\)
0.459663 + 0.888093i \(0.347970\pi\)
\(44\) −2.11713 + 6.51586i −0.319170 + 0.982303i
\(45\) 2.21490 + 0.306931i 0.330178 + 0.0457546i
\(46\) 0.750898 − 2.31103i 0.110714 − 0.340742i
\(47\) −1.25092 − 0.406447i −0.182465 − 0.0592864i 0.216359 − 0.976314i \(-0.430582\pi\)
−0.398824 + 0.917027i \(0.630582\pi\)
\(48\) −1.04514 + 1.43852i −0.150853 + 0.207632i
\(49\) 1.47441 − 4.53776i 0.210630 − 0.648252i
\(50\) 0.856166 − 3.02985i 0.121080 − 0.428486i
\(51\) −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 −0.967246 − 0.253841i \(-0.918306\pi\)
0.540313 + 0.841464i \(0.318306\pi\)
\(54\) −0.509437 − 0.370128i −0.0693256 − 0.0503680i
\(55\) 4.17165 − 8.59517i 0.562506 − 1.15897i
\(56\) 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 \(0.975405\pi\)
\(60\) 2.48571 − 2.58398i 0.320904 − 0.333591i
\(61\) −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
\(65\) −0.0804176 + 0.0143397i −0.00997458 + 0.00177862i
\(66\) −2.17667 + 1.58144i −0.267930 + 0.194662i
\(67\) 1.10338i 0.134799i 0.997726 + 0.0673995i \(0.0214702\pi\)
−0.997726 + 0.0673995i \(0.978530\pi\)
\(68\) 8.44442i 1.02404i
\(69\) −3.12193 + 2.26822i −0.375836 + 0.273061i
\(70\) −2.08216 − 0.288537i −0.248866 − 0.0344867i
\(71\) −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 \(0.107556\pi\)
0.0237357 + 0.999718i \(0.492444\pi\)
\(74\) 0.620533 + 1.90980i 0.0721355 + 0.222010i
\(75\) −3.92816 + 3.09348i −0.453584 + 0.357204i
\(76\) −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 \(0.604236\pi\)
\(80\) −0.697966 − 3.91422i −0.0780350 − 0.437623i
\(81\) 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 \(0.809273\pi\)
−0.999576 + 0.0291284i \(0.990727\pi\)
\(84\) −1.93664 − 1.40705i −0.211304 − 0.153522i
\(85\) 1.61639 − 11.6644i 0.175322 − 1.26518i
\(86\) 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 \(0.973744\pi\)
\(90\) 1.38618 − 0.247178i 0.146117 − 0.0260549i
\(91\) 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
\(95\) 0.776230 + 0.746710i 0.0796395 + 0.0766108i
\(96\) −1.74838 + 5.38097i −0.178444 + 0.549193i
\(97\) 4.70172 + 6.47137i 0.477388 + 0.657068i 0.978000 − 0.208604i \(-0.0668919\pi\)
−0.500613 + 0.865671i \(0.666892\pi\)
\(98\) − 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.4.19 yes 128
5.4 even 2 inner 465.2.ba.a.4.14 ✓ 128
31.8 even 5 inner 465.2.ba.a.349.14 yes 128
155.39 even 10 inner 465.2.ba.a.349.19 yes 128
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
465.2.ba.a.4.14 ✓ 128 5.4 even 2 inner
465.2.ba.a.4.19 yes 128 1.1 even 1 trivial
465.2.ba.a.349.14 yes 128 31.8 even 5 inner
465.2.ba.a.349.19 yes 128 155.39 even 10 inner