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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [380,2,Mod(13,380)] 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("380.13"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(380, base_ring=CyclotomicField(36)) chi = DirichletCharacter(H, H._module([0, 27, 10])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 380 = 2^{2} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 380.bh (of order \(36\), degree \(12\), 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: \(3.03431527681\)
Analytic rank: \(0\)
Dimension: \(120\)
Relative dimension: \(10\) over \(\Q(\zeta_{36})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{36}]$

Embedding invariants

Embedding label 33.2
Character \(\chi\) \(=\) 380.33
Dual form 380.2.bh.a.357.2

$q$-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)
 
\(f(q)\) \(=\) \(q+(-1.41957 - 2.02735i) q^{3} +(0.187939 - 2.22816i) q^{5} +(-3.90736 - 1.04697i) q^{7} +(-1.06893 + 2.93686i) q^{9} +(-1.50482 + 2.60643i) q^{11} +(2.49576 + 1.74755i) q^{13} +(-4.78405 + 2.78200i) q^{15} +(1.54457 - 3.31233i) q^{17} +(-2.24349 + 3.73721i) q^{19} +(3.42418 + 9.40786i) q^{21} +(4.38572 + 0.383700i) q^{23} +(-4.92936 - 0.837516i) q^{25} +(0.299637 - 0.0802874i) q^{27} +(-6.16306 - 2.24317i) q^{29} +(0.0797471 - 0.0460420i) q^{31} +(7.42036 - 0.649197i) q^{33} +(-3.06717 + 8.50944i) q^{35} +(-5.63016 - 5.63016i) q^{37} -7.54055i q^{39} +(-0.252048 - 0.0444428i) q^{41} +(-0.986904 - 11.2804i) q^{43} +(6.34288 + 2.93369i) q^{45} +(-12.1390 + 5.66049i) q^{47} +(8.10914 + 4.68182i) q^{49} +(-8.90788 + 1.57070i) q^{51} +(-0.0341343 + 0.390156i) q^{53} +(5.52472 + 3.84283i) q^{55} +(10.7614 - 0.756885i) q^{57} +(6.11253 - 2.22478i) q^{59} +(-5.92709 + 4.97342i) q^{61} +(7.25151 - 10.3562i) q^{63} +(4.36286 - 5.23251i) q^{65} +(-3.18616 - 6.83274i) q^{67} +(-5.44793 - 9.43609i) q^{69} +(10.3566 - 12.3425i) q^{71} +(4.03907 - 2.82818i) q^{73} +(5.29962 + 11.1825i) q^{75} +(8.60875 - 8.60875i) q^{77} +(1.67176 - 9.48103i) q^{79} +(6.59432 + 5.53329i) q^{81} +(-1.31821 + 4.91963i) q^{83} +(-7.09010 - 4.06405i) q^{85} +(4.20119 + 15.6790i) q^{87} +(1.38111 + 7.83264i) q^{89} +(-7.92219 - 9.44130i) q^{91} +(-0.206550 - 0.0963158i) q^{93} +(7.90545 + 5.70121i) q^{95} +(-4.48907 - 2.09329i) q^{97} +(-6.04617 - 7.20554i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 120 q + 6 q^{7} + 18 q^{15} - 18 q^{17} + 48 q^{21} - 36 q^{23} - 24 q^{25} - 60 q^{33} - 18 q^{35} - 12 q^{41} - 36 q^{43} + 18 q^{45} - 24 q^{47} + 96 q^{51} - 18 q^{53} + 72 q^{55} - 6 q^{57} - 24 q^{61}+ \cdots - 60 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/380\mathbb{Z}\right)^\times\).

\(n\) \(21\) \(77\) \(191\)
\(\chi(n)\) \(e\left(\frac{7}{18}\right)\) \(e\left(\frac{3}{4}\right)\) \(1\)

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
\(3\) −1.41957 − 2.02735i −0.819588 − 1.17049i −0.982940 − 0.183925i \(-0.941119\pi\)
0.163352 − 0.986568i \(-0.447769\pi\)
\(4\) 0 0
\(5\) 0.187939 − 2.22816i 0.0840490 − 0.996462i
\(6\) 0 0
\(7\) −3.90736 − 1.04697i −1.47684 − 0.395719i −0.571572 − 0.820552i \(-0.693666\pi\)
−0.905272 + 0.424833i \(0.860333\pi\)
\(8\) 0 0
\(9\) −1.06893 + 2.93686i −0.356310 + 0.978953i
\(10\) 0 0
\(11\) −1.50482 + 2.60643i −0.453721 + 0.785868i −0.998614 − 0.0526377i \(-0.983237\pi\)
0.544892 + 0.838506i \(0.316570\pi\)
\(12\) 0 0
\(13\) 2.49576 + 1.74755i 0.692199 + 0.484683i 0.865965 − 0.500104i \(-0.166705\pi\)
−0.173766 + 0.984787i \(0.555594\pi\)
\(14\) 0 0
\(15\) −4.78405 + 2.78200i −1.23524 + 0.718309i
\(16\) 0 0
\(17\) 1.54457 − 3.31233i 0.374612 − 0.803358i −0.625127 − 0.780523i \(-0.714953\pi\)
0.999739 − 0.0228352i \(-0.00726931\pi\)
\(18\) 0 0
\(19\) −2.24349 + 3.73721i −0.514692 + 0.857375i
\(20\) 0 0
\(21\) 3.42418 + 9.40786i 0.747217 + 2.05296i
\(22\) 0 0
\(23\) 4.38572 + 0.383700i 0.914485 + 0.0800071i 0.534675 − 0.845058i \(-0.320434\pi\)
0.379810 + 0.925065i \(0.375989\pi\)
\(24\) 0 0
\(25\) −4.92936 − 0.837516i −0.985872 − 0.167503i
\(26\) 0 0
\(27\) 0.299637 − 0.0802874i 0.0576651 − 0.0154513i
\(28\) 0 0
\(29\) −6.16306 − 2.24317i −1.14445 − 0.416546i −0.300932 − 0.953645i \(-0.597298\pi\)
−0.843519 + 0.537099i \(0.819520\pi\)
\(30\) 0 0
\(31\) 0.0797471 − 0.0460420i 0.0143230 − 0.00826939i −0.492821 − 0.870130i \(-0.664034\pi\)
0.507144 + 0.861861i \(0.330701\pi\)
\(32\) 0 0
\(33\) 7.42036 − 0.649197i 1.29172 − 0.113011i
\(34\) 0 0
\(35\) −3.06717 + 8.50944i −0.518446 + 1.43836i
\(36\) 0 0
\(37\) −5.63016 − 5.63016i −0.925592 − 0.925592i 0.0718248 − 0.997417i \(-0.477118\pi\)
−0.997417 + 0.0718248i \(0.977118\pi\)
\(38\) 0 0
\(39\) − 7.54055i − 1.20745i
\(40\) 0 0
\(41\) −0.252048 − 0.0444428i −0.0393633 − 0.00694081i 0.153932 − 0.988081i \(-0.450806\pi\)
−0.193295 + 0.981141i \(0.561917\pi\)
\(42\) 0 0
\(43\) −0.986904 − 11.2804i −0.150501 − 1.72024i −0.577842 − 0.816149i \(-0.696105\pi\)
0.427340 − 0.904091i \(-0.359451\pi\)
\(44\) 0 0
\(45\) 6.34288 + 2.93369i 0.945541 + 0.437329i
\(46\) 0 0
\(47\) −12.1390 + 5.66049i −1.77065 + 0.825667i −0.794910 + 0.606727i \(0.792482\pi\)
−0.975738 + 0.218940i \(0.929740\pi\)
\(48\) 0 0
\(49\) 8.10914 + 4.68182i 1.15845 + 0.668831i
\(50\) 0 0
\(51\) −8.90788 + 1.57070i −1.24735 + 0.219942i
\(52\) 0 0
\(53\) −0.0341343 + 0.390156i −0.00468870 + 0.0535921i −0.998179 − 0.0603282i \(-0.980785\pi\)
0.993490 + 0.113920i \(0.0363408\pi\)
\(54\) 0 0
\(55\) 5.52472 + 3.84283i 0.744953 + 0.518167i
\(56\) 0 0
\(57\) 10.7614 − 0.756885i 1.42539 − 0.100252i
\(58\) 0 0
\(59\) 6.11253 − 2.22478i 0.795784 − 0.289642i 0.0880459 − 0.996116i \(-0.471938\pi\)
0.707738 + 0.706475i \(0.249716\pi\)
\(60\) 0 0
\(61\) −5.92709 + 4.97342i −0.758886 + 0.636781i −0.937837 − 0.347077i \(-0.887174\pi\)
0.178951 + 0.983858i \(0.442730\pi\)
\(62\) 0 0
\(63\) 7.25151 − 10.3562i 0.913604 − 1.30476i
\(64\) 0 0
\(65\) 4.36286 − 5.23251i 0.541147 − 0.649013i
\(66\) 0 0
\(67\) −3.18616 − 6.83274i −0.389251 − 0.834752i −0.999107 − 0.0422399i \(-0.986551\pi\)
0.609856 − 0.792512i \(-0.291227\pi\)
\(68\) 0 0
\(69\) −5.44793 − 9.43609i −0.655853 − 1.13597i
\(70\) 0 0
\(71\) 10.3566 − 12.3425i 1.22910 − 1.46479i 0.390002 − 0.920814i \(-0.372474\pi\)
0.839102 − 0.543975i \(-0.183081\pi\)
\(72\) 0 0
\(73\) 4.03907 − 2.82818i 0.472737 − 0.331014i −0.312810 − 0.949816i \(-0.601270\pi\)
0.785547 + 0.618802i \(0.212382\pi\)
\(74\) 0 0
\(75\) 5.29962 + 11.1825i 0.611947 + 1.29124i
\(76\) 0 0
\(77\) 8.60875 − 8.60875i 0.981059 − 0.981059i
\(78\) 0 0
\(79\) 1.67176 − 9.48103i 0.188088 − 1.06670i −0.733836 − 0.679327i \(-0.762272\pi\)
0.921923 − 0.387372i \(-0.126617\pi\)
\(80\) 0 0
\(81\) 6.59432 + 5.53329i 0.732702 + 0.614810i
\(82\) 0 0
\(83\) −1.31821 + 4.91963i −0.144692 + 0.539999i 0.855077 + 0.518502i \(0.173510\pi\)
−0.999769 + 0.0214974i \(0.993157\pi\)
\(84\) 0 0
\(85\) −7.09010 − 4.06405i −0.769030 − 0.440808i
\(86\) 0 0
\(87\) 4.20119 + 15.6790i 0.450414 + 1.68097i
\(88\) 0 0
\(89\) 1.38111 + 7.83264i 0.146397 + 0.830258i 0.966235 + 0.257663i \(0.0829523\pi\)
−0.819838 + 0.572596i \(0.805937\pi\)
\(90\) 0 0
\(91\) −7.92219 − 9.44130i −0.830472 − 0.989718i
\(92\) 0 0
\(93\) −0.206550 − 0.0963158i −0.0214182 − 0.00998748i
\(94\) 0 0
\(95\) 7.90545 + 5.70121i 0.811082 + 0.584932i
\(96\) 0 0
\(97\) −4.48907 − 2.09329i −0.455796 − 0.212541i 0.181141 − 0.983457i \(-0.442021\pi\)
−0.636937 + 0.770916i \(0.719799\pi\)
\(98\) 0 0
\(99\) −6.04617 − 7.20554i −0.607663 − 0.724184i
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 380.2.bh.a.33.2 ✓ 120
5.2 odd 4 inner 380.2.bh.a.337.2 yes 120
19.15 odd 18 inner 380.2.bh.a.53.2 yes 120
95.72 even 36 inner 380.2.bh.a.357.2 yes 120
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
380.2.bh.a.33.2 ✓ 120 1.1 even 1 trivial
380.2.bh.a.53.2 yes 120 19.15 odd 18 inner
380.2.bh.a.337.2 yes 120 5.2 odd 4 inner
380.2.bh.a.357.2 yes 120 95.72 even 36 inner