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Properties

Label 580.2.z.b.241.4
Level $580$
Weight $2$
Character 580.241
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 241.4
Character \(\chi\) \(=\) 580.241
Dual form 580.2.z.b.361.4

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.327706 + 0.261337i) q^{3} +(0.900969 - 0.433884i) q^{5} +(1.39270 + 1.74638i) q^{7} +(-0.628469 + 2.75350i) q^{9} +(1.42610 - 0.325497i) q^{11} +(-0.652355 - 2.85816i) q^{13} +(-0.181863 + 0.377642i) q^{15} +6.23116i q^{17} +(-1.45944 - 1.16386i) q^{19} +(-0.912789 - 0.208338i) q^{21} +(6.25072 + 3.01019i) q^{23} +(0.623490 - 0.781831i) q^{25} +(-1.05923 - 2.19951i) q^{27} +(-4.11434 - 3.47450i) q^{29} +(4.74964 + 9.86274i) q^{31} +(-0.382276 + 0.479359i) q^{33} +(2.01250 + 0.969170i) q^{35} +(5.42785 + 1.23887i) q^{37} +(0.960722 + 0.766150i) q^{39} -2.97925i q^{41} +(-0.831503 + 1.72663i) q^{43} +(0.628469 + 2.75350i) q^{45} +(11.5310 - 2.63187i) q^{47} +(0.447387 - 1.96013i) q^{49} +(-1.62843 - 2.04199i) q^{51} +(-6.92392 + 3.33438i) q^{53} +(1.14364 - 0.912023i) q^{55} +0.782426 q^{57} +2.71380 q^{59} +(-5.89295 + 4.69947i) q^{61} +(-5.68394 + 2.73724i) q^{63} +(-1.82786 - 2.29206i) q^{65} +(-0.983560 + 4.30926i) q^{67} +(-2.83507 + 0.647087i) q^{69} +(-1.26467 - 5.54089i) q^{71} +(-5.37417 + 11.1596i) q^{73} +0.419152i q^{75} +(2.55456 + 2.03720i) q^{77} +(-3.46358 - 0.790540i) q^{79} +(-6.71192 - 3.23229i) q^{81} +(10.1505 - 12.7284i) q^{83} +(2.70360 + 5.61408i) q^{85} +(2.25631 + 0.0633856i) q^{87} +(-7.44278 - 15.4551i) q^{89} +(4.08291 - 5.11980i) q^{91} +(-4.13398 - 1.99082i) q^{93} +(-1.81989 - 0.415378i) q^{95} +(-0.784997 - 0.626014i) q^{97} +4.13132i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 36 q + 6 q^{5} - 4 q^{7} + 6 q^{9} + 4 q^{13} + 7 q^{21} - 10 q^{23} - 6 q^{25} + 63 q^{27} + 5 q^{29} + 7 q^{31} + 6 q^{33} + 4 q^{35} + 49 q^{39} - 35 q^{43} - 6 q^{45} - 28 q^{47} - 46 q^{49} - 2 q^{51}+ \cdots + 63 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(117\) \(291\) \(321\)
\(\chi(n)\) \(1\) \(1\) \(e\left(\frac{5}{14}\right)\)

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
\(3\) −0.327706 + 0.261337i −0.189201 + 0.150883i −0.713509 − 0.700646i \(-0.752895\pi\)
0.524308 + 0.851529i \(0.324324\pi\)
\(4\) 0 0
\(5\) 0.900969 − 0.433884i 0.402926 − 0.194039i
\(6\) 0 0
\(7\) 1.39270 + 1.74638i 0.526389 + 0.660071i 0.971952 − 0.235179i \(-0.0755678\pi\)
−0.445563 + 0.895251i \(0.646996\pi\)
\(8\) 0 0
\(9\) −0.628469 + 2.75350i −0.209490 + 0.917834i
\(10\) 0 0
\(11\) 1.42610 − 0.325497i 0.429984 − 0.0981411i −0.00205080 − 0.999998i \(-0.500653\pi\)
0.432035 + 0.901857i \(0.357796\pi\)
\(12\) 0 0
\(13\) −0.652355 − 2.85816i −0.180931 − 0.792710i −0.981188 − 0.193053i \(-0.938161\pi\)
0.800257 − 0.599657i \(-0.204696\pi\)
\(14\) 0 0
\(15\) −0.181863 + 0.377642i −0.0469568 + 0.0975069i
\(16\) 0 0
\(17\) 6.23116i 1.51128i 0.654988 + 0.755639i \(0.272674\pi\)
−0.654988 + 0.755639i \(0.727326\pi\)
\(18\) 0 0
\(19\) −1.45944 − 1.16386i −0.334818 − 0.267008i 0.441620 − 0.897202i \(-0.354404\pi\)
−0.776438 + 0.630194i \(0.782975\pi\)
\(20\) 0 0
\(21\) −0.912789 − 0.208338i −0.199187 − 0.0454631i
\(22\) 0 0
\(23\) 6.25072 + 3.01019i 1.30337 + 0.627668i 0.951288 − 0.308303i \(-0.0997611\pi\)
0.352078 + 0.935971i \(0.385475\pi\)
\(24\) 0 0
\(25\) 0.623490 − 0.781831i 0.124698 − 0.156366i
\(26\) 0 0
\(27\) −1.05923 − 2.19951i −0.203848 − 0.423295i
\(28\) 0 0
\(29\) −4.11434 − 3.47450i −0.764015 − 0.645199i
\(30\) 0 0
\(31\) 4.74964 + 9.86274i 0.853061 + 1.77140i 0.591699 + 0.806159i \(0.298457\pi\)
0.261362 + 0.965241i \(0.415828\pi\)
\(32\) 0 0
\(33\) −0.382276 + 0.479359i −0.0665457 + 0.0834456i
\(34\) 0 0
\(35\) 2.01250 + 0.969170i 0.340175 + 0.163820i
\(36\) 0 0
\(37\) 5.42785 + 1.23887i 0.892333 + 0.203669i 0.644018 − 0.765010i \(-0.277266\pi\)
0.248315 + 0.968679i \(0.420123\pi\)
\(38\) 0 0
\(39\) 0.960722 + 0.766150i 0.153839 + 0.122682i
\(40\) 0 0
\(41\) − 2.97925i − 0.465280i −0.972563 − 0.232640i \(-0.925264\pi\)
0.972563 − 0.232640i \(-0.0747364\pi\)
\(42\) 0 0
\(43\) −0.831503 + 1.72663i −0.126803 + 0.263309i −0.954699 − 0.297572i \(-0.903823\pi\)
0.827896 + 0.560881i \(0.189538\pi\)
\(44\) 0 0
\(45\) 0.628469 + 2.75350i 0.0936866 + 0.410468i
\(46\) 0 0
\(47\) 11.5310 − 2.63187i 1.68196 − 0.383897i 0.728415 − 0.685136i \(-0.240257\pi\)
0.953549 + 0.301239i \(0.0974002\pi\)
\(48\) 0 0
\(49\) 0.447387 − 1.96013i 0.0639124 − 0.280019i
\(50\) 0 0
\(51\) −1.62843 − 2.04199i −0.228026 − 0.285935i
\(52\) 0 0
\(53\) −6.92392 + 3.33438i −0.951073 + 0.458013i −0.844062 − 0.536245i \(-0.819842\pi\)
−0.107011 + 0.994258i \(0.534128\pi\)
\(54\) 0 0
\(55\) 1.14364 − 0.912023i 0.154208 − 0.122977i
\(56\) 0 0
\(57\) 0.782426 0.103635
\(58\) 0 0
\(59\) 2.71380 0.353306 0.176653 − 0.984273i \(-0.443473\pi\)
0.176653 + 0.984273i \(0.443473\pi\)
\(60\) 0 0
\(61\) −5.89295 + 4.69947i −0.754515 + 0.601706i −0.923359 − 0.383937i \(-0.874568\pi\)
0.168844 + 0.985643i \(0.445997\pi\)
\(62\) 0 0
\(63\) −5.68394 + 2.73724i −0.716109 + 0.344860i
\(64\) 0 0
\(65\) −1.82786 − 2.29206i −0.226718 − 0.284295i
\(66\) 0 0
\(67\) −0.983560 + 4.30926i −0.120161 + 0.526460i 0.878639 + 0.477486i \(0.158452\pi\)
−0.998800 + 0.0489732i \(0.984405\pi\)
\(68\) 0 0
\(69\) −2.83507 + 0.647087i −0.341302 + 0.0779001i
\(70\) 0 0
\(71\) −1.26467 − 5.54089i −0.150089 − 0.657583i −0.992857 − 0.119306i \(-0.961933\pi\)
0.842769 − 0.538276i \(-0.180924\pi\)
\(72\) 0 0
\(73\) −5.37417 + 11.1596i −0.628999 + 1.30613i 0.306186 + 0.951972i \(0.400947\pi\)
−0.935185 + 0.354159i \(0.884767\pi\)
\(74\) 0 0
\(75\) 0.419152i 0.0483994i
\(76\) 0 0
\(77\) 2.55456 + 2.03720i 0.291119 + 0.232160i
\(78\) 0 0
\(79\) −3.46358 − 0.790540i −0.389684 − 0.0889427i 0.0231900 − 0.999731i \(-0.492618\pi\)
−0.412874 + 0.910788i \(0.635475\pi\)
\(80\) 0 0
\(81\) −6.71192 − 3.23229i −0.745769 − 0.359144i
\(82\) 0 0
\(83\) 10.1505 − 12.7284i 1.11417 − 1.39712i 0.205979 − 0.978556i \(-0.433962\pi\)
0.908187 − 0.418564i \(-0.137466\pi\)
\(84\) 0 0
\(85\) 2.70360 + 5.61408i 0.293246 + 0.608933i
\(86\) 0 0
\(87\) 2.25631 + 0.0633856i 0.241902 + 0.00679565i
\(88\) 0 0
\(89\) −7.44278 − 15.4551i −0.788933 − 1.63824i −0.769687 − 0.638421i \(-0.779588\pi\)
−0.0192455 − 0.999815i \(-0.506126\pi\)
\(90\) 0 0
\(91\) 4.08291 − 5.11980i 0.428005 − 0.536701i
\(92\) 0 0
\(93\) −4.13398 − 1.99082i −0.428674 − 0.206438i
\(94\) 0 0
\(95\) −1.81989 − 0.415378i −0.186717 − 0.0426168i
\(96\) 0 0
\(97\) −0.784997 − 0.626014i −0.0797043 − 0.0635621i 0.582827 − 0.812596i \(-0.301946\pi\)
−0.662531 + 0.749034i \(0.730518\pi\)
\(98\) 0 0
\(99\) 4.13132i 0.415214i
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.241.4 ✓ 36
29.13 even 14 inner 580.2.z.b.361.4 yes 36
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
580.2.z.b.241.4 ✓ 36 1.1 even 1 trivial
580.2.z.b.361.4 yes 36 29.13 even 14 inner