↑

Properties

Label 465.2.ba.a.349.18
Level $465$
Weight $2$
Character 465.349
Analytic conductor $3.713$
Analytic rank $0$
Dimension $128$
Inner twists $4$

Related objects

Downloads

Learn more

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 349.18
Character \(\chi\) \(=\) 465.349
Dual form 465.2.ba.a.4.18

$q$-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 \(\left(\mathbb{Z}/465\mathbb{Z}\right)^\times\).

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

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 0.468459 − 0.883485i \(-0.344809\pi\)
−0.140308 + 0.990108i \(0.544809\pi\)
\(3\) 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 −0.630015 − 0.776583i \(-0.716951\pi\)
0.933260 + 0.359203i \(0.116951\pi\)
\(8\) −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 −0.0176207 − 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 −0.815030 − 0.579418i \(-0.803280\pi\)
−0.318800 + 0.947822i \(0.603280\pi\)
\(14\) 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 0.731184 − 0.682181i \(-0.238968\pi\)
−0.874741 + 0.484591i \(0.838968\pi\)
\(18\) 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 \(0.963845\pi\)
\(20\) 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 −0.835687 − 0.549205i \(-0.814930\pi\)
−0.264084 − 0.964500i \(-0.585070\pi\)
\(24\) −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
\(35\) −1.50426 + 2.65573i −0.254267 + 0.448900i
\(36\) −1.76190 −0.293649
\(37\) − 1.93895i − 0.318762i −0.987217 − 0.159381i \(-0.949050\pi\)
0.987217 − 0.159381i \(-0.0509498\pi\)
\(38\) −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 −0.975287 − 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 −0.494858 − 0.868974i \(-0.664780\pi\)
−0.911119 + 0.412144i \(0.864780\pi\)
\(44\) 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 0.570454 − 0.821330i \(-0.306767\pi\)
0.944272 + 0.329166i \(0.106767\pi\)
\(48\) 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 0.864870 − 0.501995i \(-0.167401\pi\)
−0.744685 + 0.667416i \(0.767401\pi\)
\(54\) 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 0.980209 − 0.197966i \(-0.0634337\pi\)
−0.909367 + 0.415994i \(0.863434\pi\)
\(60\) 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 \(0.229915\pi\)
−0.750288 + 0.661111i \(0.770085\pi\)
\(68\) 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 −0.581043 + 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 −0.415187 − 0.909736i \(-0.636284\pi\)
0.993510 + 0.113742i \(0.0362839\pi\)
\(74\) 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 −0.304092 − 0.952643i \(-0.598353\pi\)
0.812047 + 0.583592i \(0.198353\pi\)
\(80\) −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 −0.0721112 − 0.997397i \(-0.522974\pi\)
−0.644594 + 0.764525i \(0.722974\pi\)
\(84\) −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 0.348414 − 0.937341i \(-0.386720\pi\)
−0.783798 + 0.621016i \(0.786720\pi\)
\(90\) −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 −0.540280 − 0.841485i \(-0.681682\pi\)
0.967256 + 0.253803i \(0.0816817\pi\)
\(98\) 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.349.18 yes 128
5.4 even 2 inner 465.2.ba.a.349.15 yes 128
31.4 even 5 inner 465.2.ba.a.4.15 ✓ 128
155.4 even 10 inner 465.2.ba.a.4.18 yes 128
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
465.2.ba.a.4.15 ✓ 128 31.4 even 5 inner
465.2.ba.a.4.18 yes 128 155.4 even 10 inner
465.2.ba.a.349.15 yes 128 5.4 even 2 inner
465.2.ba.a.349.18 yes 128 1.1 even 1 trivial