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Properties

Label 580.2.z.b.121.4
Level $580$
Weight $2$
Character 580.121
Analytic conductor $4.631$
Analytic rank $0$
Dimension $36$
Inner twists $2$

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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] = [580,2,Mod(121,580)] 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("580.121"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(580, base_ring=CyclotomicField(14)) chi = DirichletCharacter(H, H._module([0, 0, 11])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 580 = 2^{2} \cdot 5 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 580.z (of order \(14\), degree \(6\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [36] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(4.63132331723\)
Analytic rank: \(0\)
Dimension: \(36\)
Relative dimension: \(6\) over \(\Q(\zeta_{14})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{14}]$

Embedding invariants

Embedding label 121.4
Character \(\chi\) \(=\) 580.121
Dual form 580.2.z.b.441.4

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.170057 - 0.0388144i) q^{3} +(-0.623490 - 0.781831i) q^{5} +(0.0981377 + 0.429969i) q^{7} +(-2.67549 + 1.28845i) q^{9} +(1.75892 - 3.65243i) q^{11} +(6.24370 + 3.00681i) q^{13} +(-0.136375 - 0.108756i) q^{15} -7.45985i q^{17} +(6.63724 + 1.51491i) q^{19} +(0.0333780 + 0.0693102i) q^{21} +(2.05404 - 2.57568i) q^{23} +(-0.222521 + 0.974928i) q^{25} +(-0.814102 + 0.649225i) q^{27} +(2.47369 - 4.78339i) q^{29} +(-2.87327 + 2.29136i) q^{31} +(0.157350 - 0.689394i) q^{33} +(0.274976 - 0.344809i) q^{35} +(-2.36291 - 4.90664i) q^{37} +(1.17849 + 0.268983i) q^{39} +3.17747i q^{41} +(-1.96692 - 1.56857i) q^{43} +(2.67549 + 1.28845i) q^{45} +(-3.50229 + 7.27257i) q^{47} +(6.13154 - 2.95279i) q^{49} +(-0.289550 - 1.26860i) q^{51} +(0.290612 + 0.364416i) q^{53} +(-3.95225 + 0.902076i) q^{55} +1.18751 q^{57} +9.62115 q^{59} +(-1.74530 + 0.398353i) q^{61} +(-0.816561 - 1.02393i) q^{63} +(-1.54207 - 6.75623i) q^{65} +(-8.75720 + 4.21725i) q^{67} +(0.249330 - 0.517739i) q^{69} +(13.1088 + 6.31289i) q^{71} +(-7.39152 - 5.89454i) q^{73} +0.174430i q^{75} +(1.74305 + 0.397840i) q^{77} +(0.464942 + 0.965463i) q^{79} +(5.44125 - 6.82312i) q^{81} +(-1.94260 + 8.51110i) q^{83} +(-5.83235 + 4.65114i) q^{85} +(0.235004 - 0.909465i) q^{87} +(-8.98921 + 7.16866i) q^{89} +(-0.680093 + 2.97968i) q^{91} +(-0.399683 + 0.501186i) q^{93} +(-2.95385 - 6.13374i) q^{95} +(-7.96061 - 1.81696i) q^{97} +12.0383i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 36 q + 6 q^{5} - 4 q^{7} + 6 q^{9} + 4 q^{13} + 7 q^{21} - 10 q^{23} - 6 q^{25} + 63 q^{27} + 5 q^{29} + 7 q^{31} + 6 q^{33} + 4 q^{35} + 49 q^{39} - 35 q^{43} - 6 q^{45} - 28 q^{47} - 46 q^{49} - 2 q^{51}+ \cdots + 63 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/580\mathbb{Z}\right)^\times\).

\(n\) \(117\) \(291\) \(321\)
\(\chi(n)\) \(1\) \(1\) \(e\left(\frac{11}{14}\right)\)

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 0
\(3\) 0.170057 − 0.0388144i 0.0981825 − 0.0224095i −0.173147 − 0.984896i \(-0.555394\pi\)
0.271330 + 0.962486i \(0.412537\pi\)
\(4\) 0 0
\(5\) −0.623490 − 0.781831i −0.278833 − 0.349646i
\(6\) 0 0
\(7\) 0.0981377 + 0.429969i 0.0370926 + 0.162513i 0.990082 − 0.140490i \(-0.0448678\pi\)
−0.952990 + 0.303003i \(0.902011\pi\)
\(8\) 0 0
\(9\) −2.67549 + 1.28845i −0.891831 + 0.429483i
\(10\) 0 0
\(11\) 1.75892 − 3.65243i 0.530334 − 1.10125i −0.447965 − 0.894051i \(-0.647851\pi\)
0.978299 − 0.207199i \(-0.0664346\pi\)
\(12\) 0 0
\(13\) 6.24370 + 3.00681i 1.73169 + 0.833938i 0.985811 + 0.167857i \(0.0536848\pi\)
0.745879 + 0.666081i \(0.232030\pi\)
\(14\) 0 0
\(15\) −0.136375 − 0.108756i −0.0352119 − 0.0280806i
\(16\) 0 0
\(17\) − 7.45985i − 1.80928i −0.426177 − 0.904640i \(-0.640140\pi\)
0.426177 − 0.904640i \(-0.359860\pi\)
\(18\) 0 0
\(19\) 6.63724 + 1.51491i 1.52269 + 0.347544i 0.900337 − 0.435194i \(-0.143320\pi\)
0.622352 + 0.782738i \(0.286177\pi\)
\(20\) 0 0
\(21\) 0.0333780 + 0.0693102i 0.00728369 + 0.0151247i
\(22\) 0 0
\(23\) 2.05404 − 2.57568i 0.428296 − 0.537066i −0.520121 − 0.854093i \(-0.674113\pi\)
0.948417 + 0.317027i \(0.102684\pi\)
\(24\) 0 0
\(25\) −0.222521 + 0.974928i −0.0445042 + 0.194986i
\(26\) 0 0
\(27\) −0.814102 + 0.649225i −0.156674 + 0.124943i
\(28\) 0 0
\(29\) 2.47369 − 4.78339i 0.459353 − 0.888254i
\(30\) 0 0
\(31\) −2.87327 + 2.29136i −0.516055 + 0.411540i −0.846584 − 0.532256i \(-0.821344\pi\)
0.330529 + 0.943796i \(0.392773\pi\)
\(32\) 0 0
\(33\) 0.157350 − 0.689394i 0.0273911 − 0.120008i
\(34\) 0 0
\(35\) 0.274976 − 0.344809i 0.0464794 − 0.0582833i
\(36\) 0 0
\(37\) −2.36291 − 4.90664i −0.388460 − 0.806646i −0.999882 − 0.0153415i \(-0.995116\pi\)
0.611422 − 0.791305i \(-0.290598\pi\)
\(38\) 0 0
\(39\) 1.17849 + 0.268983i 0.188710 + 0.0430718i
\(40\) 0 0
\(41\) 3.17747i 0.496238i 0.968730 + 0.248119i \(0.0798124\pi\)
−0.968730 + 0.248119i \(0.920188\pi\)
\(42\) 0 0
\(43\) −1.96692 − 1.56857i −0.299953 − 0.239204i 0.461934 − 0.886914i \(-0.347156\pi\)
−0.761887 + 0.647710i \(0.775727\pi\)
\(44\) 0 0
\(45\) 2.67549 + 1.28845i 0.398839 + 0.192071i
\(46\) 0 0
\(47\) −3.50229 + 7.27257i −0.510861 + 1.06081i 0.472864 + 0.881135i \(0.343220\pi\)
−0.983725 + 0.179679i \(0.942494\pi\)
\(48\) 0 0
\(49\) 6.13154 − 2.95279i 0.875934 − 0.421828i
\(50\) 0 0
\(51\) −0.289550 − 1.26860i −0.0405451 − 0.177640i
\(52\) 0 0
\(53\) 0.290612 + 0.364416i 0.0399186 + 0.0500563i 0.801390 − 0.598142i \(-0.204094\pi\)
−0.761471 + 0.648198i \(0.775523\pi\)
\(54\) 0 0
\(55\) −3.95225 + 0.902076i −0.532922 + 0.121636i
\(56\) 0 0
\(57\) 1.18751 0.157290
\(58\) 0 0
\(59\) 9.62115 1.25257 0.626283 − 0.779595i \(-0.284575\pi\)
0.626283 + 0.779595i \(0.284575\pi\)
\(60\) 0 0
\(61\) −1.74530 + 0.398353i −0.223463 + 0.0510039i −0.332786 − 0.943002i \(-0.607989\pi\)
0.109324 + 0.994006i \(0.465132\pi\)
\(62\) 0 0
\(63\) −0.816561 − 1.02393i −0.102877 − 0.129004i
\(64\) 0 0
\(65\) −1.54207 − 6.75623i −0.191270 − 0.838008i
\(66\) 0 0
\(67\) −8.75720 + 4.21725i −1.06986 + 0.515219i −0.884062 − 0.467369i \(-0.845202\pi\)
−0.185800 + 0.982588i \(0.559488\pi\)
\(68\) 0 0
\(69\) 0.249330 − 0.517739i 0.0300158 − 0.0623285i
\(70\) 0 0
\(71\) 13.1088 + 6.31289i 1.55573 + 0.749202i 0.996794 − 0.0800064i \(-0.0254941\pi\)
0.558940 + 0.829208i \(0.311208\pi\)
\(72\) 0 0
\(73\) −7.39152 − 5.89454i −0.865112 − 0.689904i 0.0868166 − 0.996224i \(-0.472331\pi\)
−0.951928 + 0.306321i \(0.900902\pi\)
\(74\) 0 0
\(75\) 0.174430i 0.0201415i
\(76\) 0 0
\(77\) 1.74305 + 0.397840i 0.198639 + 0.0453381i
\(78\) 0 0
\(79\) 0.464942 + 0.965463i 0.0523101 + 0.108623i 0.925485 − 0.378783i \(-0.123657\pi\)
−0.873175 + 0.487406i \(0.837943\pi\)
\(80\) 0 0
\(81\) 5.44125 − 6.82312i 0.604584 − 0.758124i
\(82\) 0 0
\(83\) −1.94260 + 8.51110i −0.213228 + 0.934215i 0.749128 + 0.662425i \(0.230473\pi\)
−0.962357 + 0.271790i \(0.912384\pi\)
\(84\) 0 0
\(85\) −5.83235 + 4.65114i −0.632607 + 0.504487i
\(86\) 0 0
\(87\) 0.235004 − 0.909465i 0.0251950 − 0.0975049i
\(88\) 0 0
\(89\) −8.98921 + 7.16866i −0.952855 + 0.759876i −0.970782 − 0.239963i \(-0.922865\pi\)
0.0179274 + 0.999839i \(0.494293\pi\)
\(90\) 0 0
\(91\) −0.680093 + 2.97968i −0.0712931 + 0.312355i
\(92\) 0 0
\(93\) −0.399683 + 0.501186i −0.0414452 + 0.0519706i
\(94\) 0 0
\(95\) −2.95385 − 6.13374i −0.303059 − 0.629308i
\(96\) 0 0
\(97\) −7.96061 − 1.81696i −0.808278 − 0.184484i −0.201637 − 0.979460i \(-0.564626\pi\)
−0.606641 + 0.794976i \(0.707483\pi\)
\(98\) 0 0
\(99\) 12.0383i 1.20990i
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 580.2.z.b.121.4 ✓ 36
29.6 even 14 inner 580.2.z.b.441.4 yes 36
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
580.2.z.b.121.4 ✓ 36 1.1 even 1 trivial
580.2.z.b.441.4 yes 36 29.6 even 14 inner