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Properties

Label 465.2.ba.a.4.18
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 4.18
Character \(\chi\) \(=\) 465.4
Dual form 465.2.ba.a.349.18

$q$-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} +(-1.06910 + 0.218057i) 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} +(-2.03519 - 0.926282i) 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} +(3.42801 - 1.94170i) 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} +(-1.50426 - 2.65573i) q^{35} -1.76190 q^{36} +1.93895i q^{37} +(-1.03721 - 1.42760i) q^{38} +(-3.47753 - 2.52657i) q^{39} +(2.77156 - 3.02762i) 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} +(-1.64934 - 1.50986i) 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} +(2.43030 - 0.215030i) 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} +(7.76868 - 4.40035i) q^{55} -2.50560 q^{56} -3.61632i q^{57} +(1.85528 + 2.55357i) q^{58} +(0.544144 - 1.67470i) q^{59} +(3.86024 - 0.787351i) q^{60} +9.09044 q^{61} +(2.66835 + 0.510965i) q^{62} +1.36497i q^{63} +(0.877286 + 2.70001i) q^{64} +(8.74819 + 3.98159i) q^{65} +(-1.57626 + 1.14522i) q^{66} -10.8229i q^{67} -1.77423i q^{68} +(-7.25948 + 5.27432i) q^{69} +(-1.09854 - 1.00564i) 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} +(4.28837 + 2.57098i) 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} +(-2.43434 + 5.34863i) 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} +(1.52043 - 1.66089i) q^{85} +(-3.82689 + 2.78040i) q^{86} +6.46856i q^{87} -7.32952i q^{88} +(-4.10740 + 2.98420i) q^{89} +(-0.993088 - 0.451987i) q^{90} +(-1.81309 - 5.58011i) q^{91} -15.8099i q^{92} +(3.80854 + 4.06141i) q^{93} +5.32794 q^{94} +(1.61605 + 7.92320i) 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 \(\left(\mathbb{Z}/465\mathbb{Z}\right)^\times\).

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

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.464076 − 0.150787i 0.328151 − 0.106623i −0.140308 − 0.990108i \(-0.544809\pi\)
0.468459 + 0.883485i \(0.344809\pi\)
\(3\) 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 \(-0.116951\pi\)
−0.630015 + 0.776583i \(0.716951\pi\)
\(8\) −1.07897 + 1.48507i −0.381472 + 0.525052i
\(9\) 0.809017 + 0.587785i 0.269672 + 0.195928i
\(10\) −1.06910 + 0.218057i −0.338078 + 0.0689557i
\(11\) −3.23031 + 2.34696i −0.973975 + 0.707634i −0.956354 − 0.292211i \(-0.905609\pi\)
−0.0176207 + 0.999845i \(0.505609\pi\)
\(12\) −1.67566 + 0.544456i −0.483722 + 0.157171i
\(13\) −4.08808 − 1.32830i −1.13383 − 0.368404i −0.318800 − 0.947822i \(-0.603280\pi\)
−0.815030 + 0.579418i \(0.803280\pi\)
\(14\) 0.538844 + 0.391493i 0.144012 + 0.104631i
\(15\) −2.03519 − 0.926282i −0.525484 − 0.239165i
\(16\) 0.812120 − 2.49945i 0.203030 − 0.624862i
\(17\) −0.591901 + 0.814682i −0.143557 + 0.197589i −0.874741 − 0.484591i \(-0.838968\pi\)
0.731184 + 0.682181i \(0.238968\pi\)
\(18\) 0.464076 + 0.150787i 0.109384 + 0.0355409i
\(19\) −1.11750 − 3.43932i −0.256373 − 0.789034i −0.993556 − 0.113341i \(-0.963845\pi\)
0.737183 − 0.675693i \(-0.236155\pi\)
\(20\) 3.42801 − 1.94170i 0.766525 − 0.434176i
\(21\) 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 −0.264084 + 0.964500i \(0.585070\pi\)
−0.835687 + 0.549205i \(0.814930\pi\)
\(24\) −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 0.946016 + 0.324119i \(0.105068\pi\)
−0.574830 + 0.818273i \(0.694932\pi\)
\(30\) −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
\(35\) −1.50426 − 2.65573i −0.254267 − 0.448900i
\(36\) −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
\(40\) 2.77156 − 3.02762i 0.438223 − 0.478708i
\(41\) −0.361110 − 1.11138i −0.0563960 − 0.173569i 0.918891 − 0.394512i \(-0.129086\pi\)
−0.975287 + 0.220943i \(0.929086\pi\)
\(42\) 0.391493 + 0.538844i 0.0604087 + 0.0831455i
\(43\) −9.21960 + 2.99563i −1.40598 + 0.456829i −0.911119 − 0.412144i \(-0.864780\pi\)
−0.494858 + 0.868974i \(0.664780\pi\)
\(44\) 2.17395 − 6.69072i 0.327735 − 1.00866i
\(45\) −1.64934 − 1.50986i −0.245870 − 0.225076i
\(46\) −1.35305 + 4.16425i −0.199496 + 0.613985i
\(47\) 10.3844 + 3.37411i 1.51473 + 0.492164i 0.944272 − 0.329166i \(-0.106767\pi\)
0.570454 + 0.821330i \(0.306767\pi\)
\(48\) 1.54474 − 2.12616i 0.222964 − 0.306884i
\(49\) 1.58738 − 4.88544i 0.226768 − 0.697920i
\(50\) 2.43030 − 0.215030i 0.343696 − 0.0304098i
\(51\) −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 \(-0.767401\pi\)
0.864870 + 0.501995i \(0.167401\pi\)
\(54\) 0.394766 + 0.286815i 0.0537209 + 0.0390305i
\(55\) 7.76868 − 4.40035i 1.04753 − 0.593343i
\(56\) −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 −0.909367 − 0.415994i \(-0.863434\pi\)
0.980209 + 0.197966i \(0.0634337\pi\)
\(60\) 3.86024 − 0.787351i 0.498355 − 0.101647i
\(61\) 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
\(65\) 8.74819 + 3.98159i 1.08508 + 0.493856i
\(66\) −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
\(70\) −1.09854 − 1.00564i −0.131301 − 0.120197i
\(71\) −12.9311 − 9.39496i −1.53463 − 1.11498i −0.953591 − 0.301104i \(-0.902645\pi\)
−0.581043 − 0.813873i \(-0.697355\pi\)
\(72\) −1.74581 + 0.567246i −0.205745 + 0.0668506i
\(73\) 4.94120 + 6.80097i 0.578323 + 0.795994i 0.993510 − 0.113742i \(-0.0362839\pi\)
−0.415187 + 0.909736i \(0.636284\pi\)
\(74\) 0.292369 + 0.899820i 0.0339872 + 0.104602i
\(75\) 4.28837 + 2.57098i 0.495178 + 0.296871i
\(76\) 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 0.812047 − 0.583592i \(-0.198353\pi\)
−0.304092 + 0.952643i \(0.598353\pi\)
\(80\) −2.43434 + 5.34863i −0.272167 + 0.597995i
\(81\) 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 \(-0.722974\pi\)
−0.0721112 + 0.997397i \(0.522974\pi\)
\(84\) −1.94563 − 1.41359i −0.212286 − 0.154235i
\(85\) 1.52043 − 1.66089i 0.164914 − 0.180149i
\(86\) −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 −0.783798 − 0.621016i \(-0.786720\pi\)
0.348414 + 0.937341i \(0.386720\pi\)
\(90\) −0.993088 − 0.451987i −0.104681 − 0.0476436i
\(91\) −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
\(95\) 1.61605 + 7.92320i 0.165803 + 0.812903i
\(96\) 1.53077 − 4.71124i 0.156234 − 0.480839i
\(97\) 4.20522 + 5.78800i 0.426976 + 0.587682i 0.967256 − 0.253803i \(-0.0816817\pi\)
−0.540280 + 0.841485i \(0.681682\pi\)
\(98\) − 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.4.18 yes 128
5.4 even 2 inner 465.2.ba.a.4.15 ✓ 128
31.8 even 5 inner 465.2.ba.a.349.15 yes 128
155.39 even 10 inner 465.2.ba.a.349.18 yes 128
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
465.2.ba.a.4.15 ✓ 128 5.4 even 2 inner
465.2.ba.a.4.18 yes 128 1.1 even 1 trivial
465.2.ba.a.349.15 yes 128 31.8 even 5 inner
465.2.ba.a.349.18 yes 128 155.39 even 10 inner