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Properties

Label 465.2.ba.a.4.7
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 4.7
Character \(\chi\) \(=\) 465.4
Dual form 465.2.ba.a.349.7

$q$-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+(-1.75841 + 0.571340i) q^{2} +(-0.951057 - 0.309017i) q^{3} +(1.14753 - 0.833726i) q^{4} +(1.89346 + 1.18945i) q^{5} +1.84890 q^{6} +(2.64500 + 3.64052i) q^{7} +(0.632036 - 0.869922i) q^{8} +(0.809017 + 0.587785i) q^{9} +(-4.00906 - 1.00972i) q^{10} +(2.28959 - 1.66349i) q^{11} +(-1.34900 + 0.438316i) q^{12} +(-2.66137 - 0.864730i) q^{13} +(-6.73095 - 4.89032i) q^{14} +(-1.43323 - 1.71635i) q^{15} +(-1.49098 + 4.58878i) q^{16} +(2.87405 - 3.95579i) q^{17} +(-1.75841 - 0.571340i) q^{18} +(-0.455764 - 1.40270i) q^{19} +(3.16447 - 0.213707i) q^{20} +(-1.39056 - 4.27969i) q^{21} +(-3.07562 + 4.23322i) q^{22} +(1.95479 - 2.69054i) q^{23} +(-0.869922 + 0.632036i) q^{24} +(2.17042 + 4.50436i) q^{25} +5.17382 q^{26} +(-0.587785 - 0.809017i) q^{27} +(6.07040 + 1.97239i) q^{28} +(2.45701 + 7.56189i) q^{29} +(3.50082 + 2.19917i) q^{30} +(1.28845 + 5.41663i) q^{31} -6.77022i q^{32} +(-2.69158 + 0.874547i) q^{33} +(-2.79364 + 8.59794i) q^{34} +(0.677986 + 10.0393i) q^{35} +1.41842 q^{36} +0.0133578i q^{37} +(1.60284 + 2.20611i) q^{38} +(2.26389 + 1.64481i) q^{39} +(2.23147 - 0.895393i) q^{40} +(-2.32425 - 7.15331i) q^{41} +(4.89032 + 6.73095i) q^{42} +(7.74137 - 2.51532i) q^{43} +(1.24047 - 3.81779i) q^{44} +(0.832704 + 2.07524i) q^{45} +(-1.90010 + 5.84792i) q^{46} +(1.87328 + 0.608664i) q^{47} +(2.83602 - 3.90345i) q^{48} +(-4.09430 + 12.6009i) q^{49} +(-6.39000 - 6.68045i) q^{50} +(-3.95579 + 2.87405i) q^{51} +(-3.77493 + 1.22655i) q^{52} +(-5.81220 + 7.99980i) q^{53} +(1.49579 + 1.08675i) q^{54} +(6.31390 - 0.426398i) q^{55} +4.83871 q^{56} +1.47488i q^{57} +(-8.64083 - 11.8931i) q^{58} +(-1.18747 + 3.65466i) q^{59} +(-3.07563 - 0.774628i) q^{60} -2.91466 q^{61} +(-5.36036 - 8.78849i) q^{62} +4.49994i q^{63} +(0.886134 + 2.72724i) q^{64} +(-4.01065 - 4.80290i) q^{65} +(4.23322 - 3.07562i) q^{66} +11.7688i q^{67} -6.93554i q^{68} +(-2.69054 + 1.95479i) q^{69} +(-6.92803 - 17.2658i) q^{70} +(-4.21032 - 3.05897i) q^{71} +(1.02266 - 0.332281i) q^{72} +(1.67005 + 2.29863i) q^{73} +(-0.00763186 - 0.0234884i) q^{74} +(-0.672267 - 4.95460i) q^{75} +(-1.69247 - 1.22965i) q^{76} +(12.1119 + 3.93540i) q^{77} +(-4.92059 - 1.59880i) q^{78} +(-1.69317 - 1.23016i) q^{79} +(-8.28125 + 6.91524i) q^{80} +(0.309017 + 0.951057i) q^{81} +(8.17395 + 11.2505i) q^{82} +(2.58759 - 0.840758i) q^{83} +(-5.16379 - 3.75171i) q^{84} +(10.1471 - 4.07161i) q^{85} +(-12.1754 + 8.84591i) q^{86} -7.95104i q^{87} -3.04315i q^{88} +(7.09866 - 5.15748i) q^{89} +(-2.64990 - 3.17335i) q^{90} +(-3.89123 - 11.9760i) q^{91} -4.71723i q^{92} +(0.448442 - 5.54968i) q^{93} -3.64173 q^{94} +(0.805465 - 3.19807i) q^{95} +(-2.09211 + 6.43886i) q^{96} +(-7.65981 - 10.5428i) q^{97} -24.4968i q^{98} +2.83009 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.75841 + 0.571340i −1.24338 + 0.403999i −0.855544 − 0.517730i \(-0.826777\pi\)
−0.387836 + 0.921728i \(0.626777\pi\)
\(3\) −0.951057 − 0.309017i −0.549093 − 0.178411i
\(4\) 1.14753 − 0.833726i 0.573763 − 0.416863i
\(5\) 1.89346 + 1.18945i 0.846783 + 0.531938i
\(6\) 1.84890 0.754809
\(7\) 2.64500 + 3.64052i 0.999714 + 1.37599i 0.925500 + 0.378747i \(0.123645\pi\)
0.0742140 + 0.997242i \(0.476355\pi\)
\(8\) 0.632036 − 0.869922i 0.223458 − 0.307564i
\(9\) 0.809017 + 0.587785i 0.269672 + 0.195928i
\(10\) −4.00906 − 1.00972i −1.26778 − 0.319302i
\(11\) 2.28959 − 1.66349i 0.690338 − 0.501560i −0.186433 − 0.982468i \(-0.559693\pi\)
0.876771 + 0.480908i \(0.159693\pi\)
\(12\) −1.34900 + 0.438316i −0.389422 + 0.126531i
\(13\) −2.66137 − 0.864730i −0.738130 − 0.239833i −0.0842642 − 0.996443i \(-0.526854\pi\)
−0.653866 + 0.756610i \(0.726854\pi\)
\(14\) −6.73095 − 4.89032i −1.79892 − 1.30699i
\(15\) −1.43323 − 1.71635i −0.370059 − 0.443159i
\(16\) −1.49098 + 4.58878i −0.372746 + 1.14719i
\(17\) 2.87405 − 3.95579i 0.697059 − 0.959420i −0.302920 − 0.953016i \(-0.597962\pi\)
0.999979 − 0.00640407i \(-0.00203849\pi\)
\(18\) −1.75841 − 0.571340i −0.414460 − 0.134666i
\(19\) −0.455764 − 1.40270i −0.104559 − 0.321801i 0.885067 − 0.465463i \(-0.154112\pi\)
−0.989627 + 0.143662i \(0.954112\pi\)
\(20\) 3.16447 − 0.213707i 0.707598 − 0.0477864i
\(21\) −1.39056 − 4.27969i −0.303444 − 0.933906i
\(22\) −3.07562 + 4.23322i −0.655723 + 0.902526i
\(23\) 1.95479 − 2.69054i 0.407603 − 0.561017i −0.555029 − 0.831831i \(-0.687293\pi\)
0.962632 + 0.270814i \(0.0872928\pi\)
\(24\) −0.869922 + 0.632036i −0.177572 + 0.129014i
\(25\) 2.17042 + 4.50436i 0.434084 + 0.900872i
\(26\) 5.17382 1.01467
\(27\) −0.587785 − 0.809017i −0.113119 − 0.155695i
\(28\) 6.07040 + 1.97239i 1.14720 + 0.372747i
\(29\) 2.45701 + 7.56189i 0.456255 + 1.40421i 0.869656 + 0.493658i \(0.164341\pi\)
−0.413401 + 0.910549i \(0.635659\pi\)
\(30\) 3.50082 + 2.19917i 0.639160 + 0.401512i
\(31\) 1.28845 + 5.41663i 0.231412 + 0.972856i
\(32\) − 6.77022i − 1.19682i
\(33\) −2.69158 + 0.874547i −0.468544 + 0.152239i
\(34\) −2.79364 + 8.59794i −0.479105 + 1.47454i
\(35\) 0.677986 + 10.0393i 0.114601 + 1.69695i
\(36\) 1.41842 0.236403
\(37\) 0.0133578i 0.00219601i 0.999999 + 0.00109801i \(0.000349506\pi\)
−0.999999 + 0.00109801i \(0.999650\pi\)
\(38\) 1.60284 + 2.20611i 0.260014 + 0.357879i
\(39\) 2.26389 + 1.64481i 0.362513 + 0.263381i
\(40\) 2.23147 − 0.895393i 0.352826 − 0.141574i
\(41\) −2.32425 − 7.15331i −0.362987 − 1.11716i −0.951232 − 0.308476i \(-0.900181\pi\)
0.588245 − 0.808683i \(-0.299819\pi\)
\(42\) 4.89032 + 6.73095i 0.754593 + 1.03861i
\(43\) 7.74137 − 2.51532i 1.18055 − 0.383583i 0.347977 − 0.937503i \(-0.386869\pi\)
0.832570 + 0.553920i \(0.186869\pi\)
\(44\) 1.24047 − 3.81779i 0.187008 − 0.575553i
\(45\) 0.832704 + 2.07524i 0.124132 + 0.309358i
\(46\) −1.90010 + 5.84792i −0.280155 + 0.862229i
\(47\) 1.87328 + 0.608664i 0.273245 + 0.0887828i 0.442435 − 0.896801i \(-0.354115\pi\)
−0.169189 + 0.985584i \(0.554115\pi\)
\(48\) 2.83602 − 3.90345i 0.409344 − 0.563414i
\(49\) −4.09430 + 12.6009i −0.584899 + 1.80014i
\(50\) −6.39000 − 6.68045i −0.903683 − 0.944758i
\(51\) −3.95579 + 2.87405i −0.553921 + 0.402447i
\(52\) −3.77493 + 1.22655i −0.523489 + 0.170092i
\(53\) −5.81220 + 7.99980i −0.798367 + 1.09886i 0.194649 + 0.980873i \(0.437643\pi\)
−0.993015 + 0.117984i \(0.962357\pi\)
\(54\) 1.49579 + 1.08675i 0.203551 + 0.147889i
\(55\) 6.31390 − 0.426398i 0.851366 − 0.0574955i
\(56\) 4.83871 0.646599
\(57\) 1.47488i 0.195353i
\(58\) −8.64083 − 11.8931i −1.13460 − 1.56164i
\(59\) −1.18747 + 3.65466i −0.154595 + 0.475796i −0.998120 − 0.0612953i \(-0.980477\pi\)
0.843524 + 0.537091i \(0.180477\pi\)
\(60\) −3.07563 − 0.774628i −0.397062 − 0.100004i
\(61\) −2.91466 −0.373184 −0.186592 − 0.982437i \(-0.559744\pi\)
−0.186592 + 0.982437i \(0.559744\pi\)
\(62\) −5.36036 − 8.78849i −0.680766 − 1.11614i
\(63\) 4.49994i 0.566939i
\(64\) 0.886134 + 2.72724i 0.110767 + 0.340905i
\(65\) −4.01065 − 4.80290i −0.497460 − 0.595726i
\(66\) 4.23322 − 3.07562i 0.521073 − 0.378582i
\(67\) 11.7688i 1.43779i 0.695119 + 0.718895i \(0.255352\pi\)
−0.695119 + 0.718895i \(0.744648\pi\)
\(68\) − 6.93554i − 0.841057i
\(69\) −2.69054 + 1.95479i −0.323903 + 0.235330i
\(70\) −6.92803 − 17.2658i −0.828058 − 2.06366i
\(71\) −4.21032 − 3.05897i −0.499673 − 0.363033i 0.309219 − 0.950991i \(-0.399932\pi\)
−0.808892 + 0.587957i \(0.799932\pi\)
\(72\) 1.02266 − 0.332281i 0.120521 − 0.0391597i
\(73\) 1.67005 + 2.29863i 0.195464 + 0.269034i 0.895488 − 0.445086i \(-0.146827\pi\)
−0.700023 + 0.714120i \(0.746827\pi\)
\(74\) −0.00763186 − 0.0234884i −0.000887186 − 0.00273048i
\(75\) −0.672267 − 4.95460i −0.0776267 − 0.572108i
\(76\) −1.69247 − 1.22965i −0.194139 − 0.141050i
\(77\) 12.1119 + 3.93540i 1.38028 + 0.448481i
\(78\) −4.92059 − 1.59880i −0.557147 − 0.181028i
\(79\) −1.69317 − 1.23016i −0.190496 − 0.138404i 0.488449 − 0.872592i \(-0.337563\pi\)
−0.678946 + 0.734189i \(0.737563\pi\)
\(80\) −8.28125 + 6.91524i −0.925871 + 0.773147i
\(81\) 0.309017 + 0.951057i 0.0343352 + 0.105673i
\(82\) 8.17395 + 11.2505i 0.902661 + 1.24241i
\(83\) 2.58759 − 0.840758i 0.284025 − 0.0922852i −0.163540 − 0.986537i \(-0.552291\pi\)
0.447565 + 0.894251i \(0.352291\pi\)
\(84\) −5.16379 − 3.75171i −0.563416 − 0.409345i
\(85\) 10.1471 − 4.07161i 1.10061 − 0.441628i
\(86\) −12.1754 + 8.84591i −1.31290 + 0.953879i
\(87\) − 7.95104i − 0.852441i
\(88\) − 3.04315i − 0.324401i
\(89\) 7.09866 − 5.15748i 0.752456 − 0.546691i −0.144131 − 0.989559i \(-0.546039\pi\)
0.896587 + 0.442867i \(0.146039\pi\)
\(90\) −2.64990 − 3.17335i −0.279324 − 0.334500i
\(91\) −3.89123 − 11.9760i −0.407912 − 1.25542i
\(92\) − 4.71723i − 0.491805i
\(93\) 0.448442 − 5.54968i 0.0465013 − 0.575475i
\(94\) −3.64173 −0.375616
\(95\) 0.805465 − 3.19807i 0.0826389 − 0.328115i
\(96\) −2.09211 + 6.43886i −0.213525 + 0.657164i
\(97\) −7.65981 − 10.5428i −0.777736 − 1.07046i −0.995528 − 0.0944670i \(-0.969885\pi\)
0.217792 − 0.975995i \(-0.430115\pi\)
\(98\) − 24.4968i − 2.47455i
\(99\) 2.83009 0.284435
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.4.7 ✓ 128
5.4 even 2 inner 465.2.ba.a.4.26 yes 128
31.8 even 5 inner 465.2.ba.a.349.26 yes 128
155.39 even 10 inner 465.2.ba.a.349.7 yes 128
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
465.2.ba.a.4.7 ✓ 128 1.1 even 1 trivial
465.2.ba.a.4.26 yes 128 5.4 even 2 inner
465.2.ba.a.349.7 yes 128 155.39 even 10 inner
465.2.ba.a.349.26 yes 128 31.8 even 5 inner