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Properties

Label 580.2.z.b.381.3
Level $580$
Weight $2$
Character 580.381
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 381.3
Character \(\chi\) \(=\) 580.381
Dual form 580.2.z.b.341.3

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.167872 - 0.348591i) q^{3} +(0.222521 - 0.974928i) q^{5} +(-3.26144 + 1.57063i) q^{7} +(1.77714 - 2.22846i) q^{9} +(1.45162 - 1.15763i) q^{11} +(0.368560 + 0.462160i) q^{13} +(-0.377206 + 0.0860948i) q^{15} -8.00923i q^{17} +(-1.18111 + 2.45259i) q^{19} +(1.09501 + 0.873243i) q^{21} +(-1.73724 - 7.61135i) q^{23} +(-0.900969 - 0.433884i) q^{25} +(-2.20677 - 0.503680i) q^{27} +(1.05677 - 5.28046i) q^{29} +(2.18047 + 0.497678i) q^{31} +(-0.647225 - 0.311687i) q^{33} +(0.805510 + 3.52917i) q^{35} +(-4.84421 - 3.86313i) q^{37} +(0.0992335 - 0.206061i) q^{39} +5.58529i q^{41} +(5.23544 - 1.19496i) q^{43} +(-1.77714 - 2.22846i) q^{45} +(-6.24072 + 4.97681i) q^{47} +(3.80570 - 4.77220i) q^{49} +(-2.79194 + 1.34453i) q^{51} +(-1.34454 + 5.89083i) q^{53} +(-0.805588 - 1.67282i) q^{55} +1.05322 q^{57} +9.69244 q^{59} +(-3.82453 - 7.94172i) q^{61} +(-2.29595 + 10.0592i) q^{63} +(0.532585 - 0.256479i) q^{65} +(2.64315 - 3.31441i) q^{67} +(-2.36161 + 1.88332i) q^{69} +(-4.77556 - 5.98837i) q^{71} +(9.26529 - 2.11474i) q^{73} +0.386906i q^{75} +(-2.91617 + 6.05549i) q^{77} +(4.01510 + 3.20193i) q^{79} +(-1.70788 - 7.48271i) q^{81} +(1.33178 + 0.641351i) q^{83} +(-7.80842 - 1.78222i) q^{85} +(-2.01812 + 0.518064i) q^{87} +(15.4840 + 3.53412i) q^{89} +(-1.92792 - 0.928437i) q^{91} +(-0.192555 - 0.843638i) q^{93} +(2.12828 + 1.69725i) q^{95} +(-4.85036 + 10.0719i) q^{97} -5.29214i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 36 q + 6 q^{5} - 4 q^{7} + 6 q^{9} + 4 q^{13} + 7 q^{21} - 10 q^{23} - 6 q^{25} + 63 q^{27} + 5 q^{29} + 7 q^{31} + 6 q^{33} + 4 q^{35} + 49 q^{39} - 35 q^{43} - 6 q^{45} - 28 q^{47} - 46 q^{49} - 2 q^{51}+ \cdots + 63 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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
\(3\) −0.167872 − 0.348591i −0.0969212 − 0.201259i 0.846871 − 0.531798i \(-0.178484\pi\)
−0.943792 + 0.330540i \(0.892769\pi\)
\(4\) 0 0
\(5\) 0.222521 − 0.974928i 0.0995144 − 0.436001i
\(6\) 0 0
\(7\) −3.26144 + 1.57063i −1.23271 + 0.593641i −0.932824 − 0.360334i \(-0.882663\pi\)
−0.299886 + 0.953975i \(0.596949\pi\)
\(8\) 0 0
\(9\) 1.77714 − 2.22846i 0.592378 − 0.742819i
\(10\) 0 0
\(11\) 1.45162 − 1.15763i 0.437680 − 0.349038i −0.379727 − 0.925099i \(-0.623982\pi\)
0.817407 + 0.576060i \(0.195411\pi\)
\(12\) 0 0
\(13\) 0.368560 + 0.462160i 0.102220 + 0.128180i 0.830310 − 0.557301i \(-0.188163\pi\)
−0.728090 + 0.685481i \(0.759592\pi\)
\(14\) 0 0
\(15\) −0.377206 + 0.0860948i −0.0973941 + 0.0222296i
\(16\) 0 0
\(17\) − 8.00923i − 1.94252i −0.238015 − 0.971261i \(-0.576497\pi\)
0.238015 − 0.971261i \(-0.423503\pi\)
\(18\) 0 0
\(19\) −1.18111 + 2.45259i −0.270964 + 0.562663i −0.991401 − 0.130858i \(-0.958227\pi\)
0.720437 + 0.693520i \(0.243941\pi\)
\(20\) 0 0
\(21\) 1.09501 + 0.873243i 0.238951 + 0.190557i
\(22\) 0 0
\(23\) −1.73724 − 7.61135i −0.362240 − 1.58708i −0.747495 − 0.664267i \(-0.768744\pi\)
0.385255 − 0.922810i \(-0.374113\pi\)
\(24\) 0 0
\(25\) −0.900969 − 0.433884i −0.180194 − 0.0867767i
\(26\) 0 0
\(27\) −2.20677 − 0.503680i −0.424693 − 0.0969334i
\(28\) 0 0
\(29\) 1.05677 − 5.28046i 0.196237 − 0.980557i
\(30\) 0 0
\(31\) 2.18047 + 0.497678i 0.391624 + 0.0893856i 0.413798 − 0.910369i \(-0.364202\pi\)
−0.0221746 + 0.999754i \(0.507059\pi\)
\(32\) 0 0
\(33\) −0.647225 − 0.311687i −0.112667 − 0.0542578i
\(34\) 0 0
\(35\) 0.805510 + 3.52917i 0.136156 + 0.596538i
\(36\) 0 0
\(37\) −4.84421 − 3.86313i −0.796384 − 0.635095i 0.138374 − 0.990380i \(-0.455812\pi\)
−0.934758 + 0.355285i \(0.884384\pi\)
\(38\) 0 0
\(39\) 0.0992335 − 0.206061i 0.0158901 − 0.0329961i
\(40\) 0 0
\(41\) 5.58529i 0.872275i 0.899880 + 0.436138i \(0.143654\pi\)
−0.899880 + 0.436138i \(0.856346\pi\)
\(42\) 0 0
\(43\) 5.23544 − 1.19496i 0.798397 − 0.182229i 0.196187 − 0.980567i \(-0.437144\pi\)
0.602210 + 0.798338i \(0.294287\pi\)
\(44\) 0 0
\(45\) −1.77714 − 2.22846i −0.264920 − 0.332199i
\(46\) 0 0
\(47\) −6.24072 + 4.97681i −0.910303 + 0.725943i −0.962095 − 0.272715i \(-0.912078\pi\)
0.0517916 + 0.998658i \(0.483507\pi\)
\(48\) 0 0
\(49\) 3.80570 − 4.77220i 0.543672 − 0.681743i
\(50\) 0 0
\(51\) −2.79194 + 1.34453i −0.390950 + 0.188272i
\(52\) 0 0
\(53\) −1.34454 + 5.89083i −0.184687 + 0.809168i 0.794672 + 0.607040i \(0.207643\pi\)
−0.979359 + 0.202129i \(0.935214\pi\)
\(54\) 0 0
\(55\) −0.805588 − 1.67282i −0.108626 − 0.225563i
\(56\) 0 0
\(57\) 1.05322 0.139503
\(58\) 0 0
\(59\) 9.69244 1.26185 0.630924 − 0.775845i \(-0.282676\pi\)
0.630924 + 0.775845i \(0.282676\pi\)
\(60\) 0 0
\(61\) −3.82453 − 7.94172i −0.489681 − 1.01683i −0.988653 − 0.150215i \(-0.952003\pi\)
0.498973 − 0.866618i \(-0.333711\pi\)
\(62\) 0 0
\(63\) −2.29595 + 10.0592i −0.289262 + 1.26734i
\(64\) 0 0
\(65\) 0.532585 − 0.256479i 0.0660590 − 0.0318124i
\(66\) 0 0
\(67\) 2.64315 − 3.31441i 0.322913 − 0.404920i −0.593706 − 0.804682i \(-0.702336\pi\)
0.916619 + 0.399762i \(0.130907\pi\)
\(68\) 0 0
\(69\) −2.36161 + 1.88332i −0.284305 + 0.226725i
\(70\) 0 0
\(71\) −4.77556 − 5.98837i −0.566755 − 0.710688i 0.413035 − 0.910715i \(-0.364469\pi\)
−0.979790 + 0.200027i \(0.935897\pi\)
\(72\) 0 0
\(73\) 9.26529 − 2.11474i 1.08442 − 0.247512i 0.357278 − 0.933998i \(-0.383705\pi\)
0.727143 + 0.686486i \(0.240848\pi\)
\(74\) 0 0
\(75\) 0.386906i 0.0446761i
\(76\) 0 0
\(77\) −2.91617 + 6.05549i −0.332329 + 0.690087i
\(78\) 0 0
\(79\) 4.01510 + 3.20193i 0.451734 + 0.360245i 0.822772 − 0.568372i \(-0.192427\pi\)
−0.371038 + 0.928618i \(0.620998\pi\)
\(80\) 0 0
\(81\) −1.70788 − 7.48271i −0.189764 − 0.831412i
\(82\) 0 0
\(83\) 1.33178 + 0.641351i 0.146182 + 0.0703975i 0.505545 − 0.862801i \(-0.331292\pi\)
−0.359363 + 0.933198i \(0.617006\pi\)
\(84\) 0 0
\(85\) −7.80842 − 1.78222i −0.846942 − 0.193309i
\(86\) 0 0
\(87\) −2.01812 + 0.518064i −0.216365 + 0.0555423i
\(88\) 0 0
\(89\) 15.4840 + 3.53412i 1.64130 + 0.374616i 0.940780 − 0.339018i \(-0.110095\pi\)
0.700519 + 0.713634i \(0.252952\pi\)
\(90\) 0 0
\(91\) −1.92792 − 0.928437i −0.202101 − 0.0973266i
\(92\) 0 0
\(93\) −0.192555 − 0.843638i −0.0199670 − 0.0874811i
\(94\) 0 0
\(95\) 2.12828 + 1.69725i 0.218357 + 0.174134i
\(96\) 0 0
\(97\) −4.85036 + 10.0719i −0.492479 + 1.02264i 0.495579 + 0.868563i \(0.334956\pi\)
−0.988058 + 0.154080i \(0.950759\pi\)
\(98\) 0 0
\(99\) − 5.29214i − 0.531880i
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.381.3 yes 36
29.22 even 14 inner 580.2.z.b.341.3 ✓ 36
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
580.2.z.b.341.3 ✓ 36 29.22 even 14 inner
580.2.z.b.381.3 yes 36 1.1 even 1 trivial