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Properties

Label 465.2.ba.a.349.15
Level $465$
Weight $2$
Character 465.349
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 349.15
Character \(\chi\) \(=\) 465.349
Dual form 465.2.ba.a.4.15

$q$-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+(-0.464076 - 0.150787i) q^{2} +(-0.951057 + 0.309017i) q^{3} +(-1.42540 - 1.03562i) q^{4} +(-2.22182 - 0.252038i) q^{5} +0.487958 q^{6} +(-0.802309 + 1.10428i) q^{7} +(1.07897 + 1.48507i) q^{8} +(0.809017 - 0.587785i) q^{9} +(0.993088 + 0.451987i) q^{10} +(-3.23031 - 2.34696i) q^{11} +(1.67566 + 0.544456i) q^{12} +(4.08808 - 1.32830i) q^{13} +(0.538844 - 0.391493i) q^{14} +(2.19096 - 0.446877i) q^{15} +(0.812120 + 2.49945i) q^{16} +(0.591901 + 0.814682i) q^{17} +(-0.464076 + 0.150787i) q^{18} +(-1.11750 + 3.43932i) q^{19} +(2.90597 + 2.66021i) q^{20} +(0.421799 - 1.29816i) q^{21} +(1.14522 + 1.57626i) q^{22} +(5.27432 + 7.25948i) q^{23} +(-1.48507 - 1.07897i) q^{24} +(4.87295 + 1.11997i) q^{25} -2.09747 q^{26} +(-0.587785 + 0.809017i) q^{27} +(2.28723 - 0.743166i) q^{28} +(1.99890 - 6.15197i) q^{29} +(-1.08415 - 0.122984i) q^{30} +(4.87718 - 2.68573i) q^{31} -4.95369i q^{32} +(3.79746 + 1.23387i) q^{33} +(-0.151843 - 0.467325i) q^{34} +(2.06091 - 2.25131i) q^{35} -1.76190 q^{36} +1.93895i q^{37} +(1.03721 - 1.42760i) q^{38} +(-3.47753 + 2.52657i) q^{39} +(-2.02297 - 3.57150i) q^{40} +(-0.361110 + 1.11138i) q^{41} +(-0.391493 + 0.538844i) q^{42} +(9.21960 + 2.99563i) q^{43} +(2.17395 + 6.69072i) q^{44} +(-1.94563 + 1.10205i) q^{45} +(-1.35305 - 4.16425i) q^{46} +(-10.3844 + 3.37411i) q^{47} +(-1.54474 - 2.12616i) q^{48} +(1.58738 + 4.88544i) q^{49} +(-2.09254 - 1.25453i) q^{50} +(-0.814682 - 0.591901i) q^{51} +(-7.20278 - 2.34032i) q^{52} +(-0.874959 - 1.20428i) q^{53} +(0.394766 - 0.286815i) q^{54} +(6.58564 + 6.02867i) q^{55} -2.50560 q^{56} -3.61632i q^{57} +(-1.85528 + 2.55357i) q^{58} +(0.544144 + 1.67470i) q^{59} +(-3.58580 - 1.63201i) q^{60} +9.09044 q^{61} +(-2.66835 + 0.510965i) q^{62} +1.36497i q^{63} +(0.877286 - 2.70001i) q^{64} +(-9.41776 + 1.92088i) q^{65} +(-1.57626 - 1.14522i) q^{66} -10.8229i q^{67} -1.77423i q^{68} +(-7.25948 - 5.27432i) q^{69} +(-1.29589 + 0.734018i) q^{70} +(-12.9311 + 9.39496i) q^{71} +(1.74581 + 0.567246i) q^{72} +(-4.94120 + 6.80097i) q^{73} +(0.292369 - 0.899820i) q^{74} +(-4.98054 + 0.440673i) q^{75} +(5.15471 - 3.74512i) q^{76} +(5.18341 - 1.68419i) q^{77} +(1.99481 - 0.648154i) q^{78} +(4.51480 - 3.28020i) q^{79} +(-1.17443 - 5.75800i) q^{80} +(0.309017 - 0.951057i) q^{81} +(0.335165 - 0.461315i) q^{82} +(6.52950 + 2.12156i) q^{83} +(-1.94563 + 1.41359i) q^{84} +(-1.10977 - 1.95926i) q^{85} +(-3.82689 - 2.78040i) q^{86} +6.46856i q^{87} -7.32952i q^{88} +(-4.10740 - 2.98420i) q^{89} +(1.06910 - 0.218057i) q^{90} +(-1.81309 + 5.58011i) q^{91} -15.8099i q^{92} +(-3.80854 + 4.06141i) q^{93} +5.32794 q^{94} +(3.34973 - 7.35989i) q^{95} +(1.53077 + 4.71124i) q^{96} +(-4.20522 + 5.78800i) q^{97} -2.50657i q^{98} -3.99288 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 \(\left(\mathbb{Z}/465\mathbb{Z}\right)^\times\).

\(n\) \(187\) \(311\) \(406\)
\(\chi(n)\) \(-1\) \(1\) \(e\left(\frac{2}{5}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.464076 − 0.150787i −0.328151 − 0.106623i 0.140308 − 0.990108i \(-0.455191\pi\)
−0.468459 + 0.883485i \(0.655191\pi\)
\(3\) −0.951057 + 0.309017i −0.549093 + 0.178411i
\(4\) −1.42540 − 1.03562i −0.712702 − 0.517808i
\(5\) −2.22182 − 0.252038i −0.993627 − 0.112715i
\(6\) 0.487958 0.199208
\(7\) −0.802309 + 1.10428i −0.303244 + 0.417380i −0.933260 − 0.359203i \(-0.883049\pi\)
0.630015 + 0.776583i \(0.283049\pi\)
\(8\) 1.07897 + 1.48507i 0.381472 + 0.525052i
\(9\) 0.809017 − 0.587785i 0.269672 − 0.195928i
\(10\) 0.993088 + 0.451987i 0.314042 + 0.142931i
\(11\) −3.23031 − 2.34696i −0.973975 − 0.707634i −0.0176207 − 0.999845i \(-0.505609\pi\)
−0.956354 + 0.292211i \(0.905609\pi\)
\(12\) 1.67566 + 0.544456i 0.483722 + 0.157171i
\(13\) 4.08808 − 1.32830i 1.13383 − 0.368404i 0.318800 − 0.947822i \(-0.396720\pi\)
0.815030 + 0.579418i \(0.196720\pi\)
\(14\) 0.538844 − 0.391493i 0.144012 − 0.104631i
\(15\) 2.19096 − 0.446877i 0.565703 − 0.115383i
\(16\) 0.812120 + 2.49945i 0.203030 + 0.624862i
\(17\) 0.591901 + 0.814682i 0.143557 + 0.197589i 0.874741 − 0.484591i \(-0.161032\pi\)
−0.731184 + 0.682181i \(0.761032\pi\)
\(18\) −0.464076 + 0.150787i −0.109384 + 0.0355409i
\(19\) −1.11750 + 3.43932i −0.256373 + 0.789034i 0.737183 + 0.675693i \(0.236155\pi\)
−0.993556 + 0.113341i \(0.963845\pi\)
\(20\) 2.90597 + 2.66021i 0.649796 + 0.594841i
\(21\) 0.421799 − 1.29816i 0.0920441 − 0.283283i
\(22\) 1.14522 + 1.57626i 0.244161 + 0.336059i
\(23\) 5.27432 + 7.25948i 1.09977 + 1.51371i 0.835687 + 0.549205i \(0.185070\pi\)
0.264084 + 0.964500i \(0.414930\pi\)
\(24\) −1.48507 − 1.07897i −0.303139 − 0.220243i
\(25\) 4.87295 + 1.11997i 0.974591 + 0.223993i
\(26\) −2.09747 −0.411348
\(27\) −0.587785 + 0.809017i −0.113119 + 0.155695i
\(28\) 2.28723 − 0.743166i 0.432246 − 0.140445i
\(29\) 1.99890 − 6.15197i 0.371186 − 1.14239i −0.574830 − 0.818273i \(-0.694932\pi\)
0.946016 − 0.324119i \(-0.105068\pi\)
\(30\) −1.08415 − 0.122984i −0.197939 − 0.0224537i
\(31\) 4.87718 − 2.68573i 0.875967 − 0.482371i
\(32\) − 4.95369i − 0.875696i
\(33\) 3.79746 + 1.23387i 0.661052 + 0.214789i
\(34\) −0.151843 − 0.467325i −0.0260409 − 0.0801456i
\(35\) 2.06091 − 2.25131i 0.348357 − 0.380540i
\(36\) −1.76190 −0.293649
\(37\) 1.93895i 0.318762i 0.987217 + 0.159381i \(0.0509498\pi\)
−0.987217 + 0.159381i \(0.949050\pi\)
\(38\) 1.03721 − 1.42760i 0.168258 − 0.231587i
\(39\) −3.47753 + 2.52657i −0.556850 + 0.404576i
\(40\) −2.02297 − 3.57150i −0.319860 − 0.564703i
\(41\) −0.361110 + 1.11138i −0.0563960 + 0.173569i −0.975287 − 0.220943i \(-0.929086\pi\)
0.918891 + 0.394512i \(0.129086\pi\)
\(42\) −0.391493 + 0.538844i −0.0604087 + 0.0831455i
\(43\) 9.21960 + 2.99563i 1.40598 + 0.456829i 0.911119 − 0.412144i \(-0.135220\pi\)
0.494858 + 0.868974i \(0.335220\pi\)
\(44\) 2.17395 + 6.69072i 0.327735 + 1.00866i
\(45\) −1.94563 + 1.10205i −0.290038 + 0.164284i
\(46\) −1.35305 − 4.16425i −0.199496 − 0.613985i
\(47\) −10.3844 + 3.37411i −1.51473 + 0.492164i −0.944272 − 0.329166i \(-0.893233\pi\)
−0.570454 + 0.821330i \(0.693233\pi\)
\(48\) −1.54474 − 2.12616i −0.222964 − 0.306884i
\(49\) 1.58738 + 4.88544i 0.226768 + 0.697920i
\(50\) −2.09254 − 1.25453i −0.295930 − 0.177417i
\(51\) −0.814682 − 0.591901i −0.114078 − 0.0828827i
\(52\) −7.20278 − 2.34032i −0.998846 − 0.324545i
\(53\) −0.874959 − 1.20428i −0.120185 − 0.165420i 0.744685 − 0.667416i \(-0.232599\pi\)
−0.864870 + 0.501995i \(0.832599\pi\)
\(54\) 0.394766 − 0.286815i 0.0537209 − 0.0390305i
\(55\) 6.58564 + 6.02867i 0.888007 + 0.812906i
\(56\) −2.50560 −0.334826
\(57\) − 3.61632i − 0.478993i
\(58\) −1.85528 + 2.55357i −0.243610 + 0.335300i
\(59\) 0.544144 + 1.67470i 0.0708416 + 0.218028i 0.980209 − 0.197966i \(-0.0634337\pi\)
−0.909367 + 0.415994i \(0.863434\pi\)
\(60\) −3.58580 − 1.63201i −0.462924 − 0.210692i
\(61\) 9.09044 1.16391 0.581956 − 0.813220i \(-0.302288\pi\)
0.581956 + 0.813220i \(0.302288\pi\)
\(62\) −2.66835 + 0.510965i −0.338881 + 0.0648927i
\(63\) 1.36497i 0.171970i
\(64\) 0.877286 − 2.70001i 0.109661 − 0.337501i
\(65\) −9.41776 + 1.92088i −1.16813 + 0.238256i
\(66\) −1.57626 − 1.14522i −0.194024 − 0.140966i
\(67\) − 10.8229i − 1.32222i −0.750288 − 0.661111i \(-0.770085\pi\)
0.750288 − 0.661111i \(-0.229915\pi\)
\(68\) − 1.77423i − 0.215157i
\(69\) −7.25948 − 5.27432i −0.873938 − 0.634953i
\(70\) −1.29589 + 0.734018i −0.154888 + 0.0877319i
\(71\) −12.9311 + 9.39496i −1.53463 + 1.11498i −0.581043 + 0.813873i \(0.697355\pi\)
−0.953591 + 0.301104i \(0.902645\pi\)
\(72\) 1.74581 + 0.567246i 0.205745 + 0.0668506i
\(73\) −4.94120 + 6.80097i −0.578323 + 0.795994i −0.993510 − 0.113742i \(-0.963716\pi\)
0.415187 + 0.909736i \(0.363716\pi\)
\(74\) 0.292369 − 0.899820i 0.0339872 − 0.104602i
\(75\) −4.98054 + 0.440673i −0.575104 + 0.0508846i
\(76\) 5.15471 − 3.74512i 0.591286 − 0.429594i
\(77\) 5.18341 − 1.68419i 0.590705 − 0.191932i
\(78\) 1.99481 − 0.648154i 0.225868 − 0.0733890i
\(79\) 4.51480 − 3.28020i 0.507955 − 0.369051i −0.304092 − 0.952643i \(-0.598353\pi\)
0.812047 + 0.583592i \(0.198353\pi\)
\(80\) −1.17443 − 5.75800i −0.131305 − 0.643764i
\(81\) 0.309017 − 0.951057i 0.0343352 − 0.105673i
\(82\) 0.335165 − 0.461315i 0.0370128 − 0.0509437i
\(83\) 6.52950 + 2.12156i 0.716705 + 0.232872i 0.644594 − 0.764525i \(-0.277026\pi\)
0.0721112 + 0.997397i \(0.477026\pi\)
\(84\) −1.94563 + 1.41359i −0.212286 + 0.154235i
\(85\) −1.10977 − 1.95926i −0.120371 − 0.212511i
\(86\) −3.82689 − 2.78040i −0.412664 − 0.299818i
\(87\) 6.46856i 0.693503i
\(88\) − 7.32952i − 0.781330i
\(89\) −4.10740 − 2.98420i −0.435384 − 0.316325i 0.348414 − 0.937341i \(-0.386720\pi\)
−0.783798 + 0.621016i \(0.786720\pi\)
\(90\) 1.06910 − 0.218057i 0.112693 − 0.0229852i
\(91\) −1.81309 + 5.58011i −0.190063 + 0.584954i
\(92\) − 15.8099i − 1.64829i
\(93\) −3.80854 + 4.06141i −0.394927 + 0.421149i
\(94\) 5.32794 0.549535
\(95\) 3.34973 − 7.35989i 0.343675 − 0.755109i
\(96\) 1.53077 + 4.71124i 0.156234 + 0.480839i
\(97\) −4.20522 + 5.78800i −0.426976 + 0.587682i −0.967256 − 0.253803i \(-0.918318\pi\)
0.540280 + 0.841485i \(0.318318\pi\)
\(98\) − 2.50657i − 0.253202i
\(99\) −3.99288 −0.401300
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.349.15 yes 128
5.4 even 2 inner 465.2.ba.a.349.18 yes 128
31.4 even 5 inner 465.2.ba.a.4.18 yes 128
155.4 even 10 inner 465.2.ba.a.4.15 ✓ 128
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
465.2.ba.a.4.15 ✓ 128 155.4 even 10 inner
465.2.ba.a.4.18 yes 128 31.4 even 5 inner
465.2.ba.a.349.15 yes 128 1.1 even 1 trivial
465.2.ba.a.349.18 yes 128 5.4 even 2 inner