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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 317.4
Character \(\chi\) \(=\) 380.317
Dual form 380.2.bh.a.193.4

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.447324 - 0.959290i) q^{3} +(-0.943399 + 2.02731i) q^{5} +(-0.763694 + 0.204631i) q^{7} +(1.20823 - 1.43991i) q^{9} +(2.14248 + 3.71088i) q^{11} +(2.61133 + 1.21768i) q^{13} +(2.36679 - 0.00187315i) q^{15} +(6.22409 + 0.544538i) q^{17} +(-0.761826 + 4.29181i) q^{19} +(0.537919 + 0.641067i) q^{21} +(-1.87517 - 2.67802i) q^{23} +(-3.22000 - 3.82513i) q^{25} +(-4.98893 - 1.33678i) q^{27} +(6.38036 + 5.35376i) q^{29} +(4.23651 + 2.44595i) q^{31} +(2.60143 - 3.71522i) q^{33} +(0.305617 - 1.74130i) q^{35} +(1.75916 - 1.75916i) q^{37} -3.04972i q^{39} +(0.896173 - 2.46221i) q^{41} +(-7.71890 - 5.40483i) q^{43} +(1.77930 + 3.80786i) q^{45} +(0.378221 + 4.32309i) q^{47} +(-5.52082 + 3.18745i) q^{49} +(-2.26182 - 6.21429i) q^{51} +(-0.408759 + 0.286216i) q^{53} +(-9.54433 + 0.842633i) q^{55} +(4.45787 - 1.18902i) q^{57} +(9.10551 - 7.64043i) q^{59} +(-1.00870 + 5.72060i) q^{61} +(-0.628064 + 1.34689i) q^{63} +(-4.93214 + 4.14522i) q^{65} +(4.30306 - 0.376469i) q^{67} +(-1.73019 + 2.99678i) q^{69} +(-15.2125 + 2.68238i) q^{71} +(-12.8258 + 5.98076i) q^{73} +(-2.22903 + 4.79998i) q^{75} +(-2.39556 - 2.39556i) q^{77} +(-1.72702 - 0.628586i) q^{79} +(-0.0298904 - 0.169517i) q^{81} +(-2.44614 - 9.12912i) q^{83} +(-6.97575 + 12.1045i) q^{85} +(2.28172 - 8.51548i) q^{87} +(12.0968 - 4.40288i) q^{89} +(-2.24343 - 0.395577i) q^{91} +(0.451282 - 5.15817i) q^{93} +(-7.98213 - 5.59335i) q^{95} +(1.34195 - 15.3386i) q^{97} +(7.93192 + 1.39861i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 120 q + 6 q^{7} + 18 q^{15} - 18 q^{17} + 48 q^{21} - 36 q^{23} - 24 q^{25} - 60 q^{33} - 18 q^{35} - 12 q^{41} - 36 q^{43} + 18 q^{45} - 24 q^{47} + 96 q^{51} - 18 q^{53} + 72 q^{55} - 6 q^{57} - 24 q^{61}+ \cdots - 60 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(21\) \(77\) \(191\)
\(\chi(n)\) \(e\left(\frac{5}{18}\right)\) \(e\left(\frac{1}{4}\right)\) \(1\)

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
\(3\) −0.447324 − 0.959290i −0.258263 − 0.553846i 0.733833 − 0.679330i \(-0.237729\pi\)
−0.992096 + 0.125484i \(0.959952\pi\)
\(4\) 0 0
\(5\) −0.943399 + 2.02731i −0.421901 + 0.906642i
\(6\) 0 0
\(7\) −0.763694 + 0.204631i −0.288649 + 0.0773433i −0.400238 − 0.916411i \(-0.631073\pi\)
0.111589 + 0.993754i \(0.464406\pi\)
\(8\) 0 0
\(9\) 1.20823 − 1.43991i 0.402742 − 0.479969i
\(10\) 0 0
\(11\) 2.14248 + 3.71088i 0.645982 + 1.11887i 0.984074 + 0.177760i \(0.0568851\pi\)
−0.338092 + 0.941113i \(0.609782\pi\)
\(12\) 0 0
\(13\) 2.61133 + 1.21768i 0.724252 + 0.337724i 0.749543 − 0.661956i \(-0.230273\pi\)
−0.0252910 + 0.999680i \(0.508051\pi\)
\(14\) 0 0
\(15\) 2.36679 − 0.00187315i 0.611101 − 0.000483645i
\(16\) 0 0
\(17\) 6.22409 + 0.544538i 1.50956 + 0.132070i 0.811664 − 0.584125i \(-0.198562\pi\)
0.697900 + 0.716195i \(0.254118\pi\)
\(18\) 0 0
\(19\) −0.761826 + 4.29181i −0.174775 + 0.984608i
\(20\) 0 0
\(21\) 0.537919 + 0.641067i 0.117384 + 0.139892i
\(22\) 0 0
\(23\) −1.87517 − 2.67802i −0.391000 − 0.558406i 0.574601 − 0.818434i \(-0.305157\pi\)
−0.965601 + 0.260028i \(0.916268\pi\)
\(24\) 0 0
\(25\) −3.22000 − 3.82513i −0.643999 − 0.765026i
\(26\) 0 0
\(27\) −4.98893 − 1.33678i −0.960121 − 0.257264i
\(28\) 0 0
\(29\) 6.38036 + 5.35376i 1.18480 + 0.994168i 0.999935 + 0.0114049i \(0.00363038\pi\)
0.184869 + 0.982763i \(0.440814\pi\)
\(30\) 0 0
\(31\) 4.23651 + 2.44595i 0.760900 + 0.439306i 0.829619 − 0.558330i \(-0.188558\pi\)
−0.0687188 + 0.997636i \(0.521891\pi\)
\(32\) 0 0
\(33\) 2.60143 − 3.71522i 0.452851 − 0.646738i
\(34\) 0 0
\(35\) 0.305617 − 1.74130i 0.0516586 − 0.294333i
\(36\) 0 0
\(37\) 1.75916 − 1.75916i 0.289204 − 0.289204i −0.547561 − 0.836766i \(-0.684444\pi\)
0.836766 + 0.547561i \(0.184444\pi\)
\(38\) 0 0
\(39\) − 3.04972i − 0.488346i
\(40\) 0 0
\(41\) 0.896173 − 2.46221i 0.139959 − 0.384533i −0.849834 − 0.527051i \(-0.823298\pi\)
0.989792 + 0.142518i \(0.0455199\pi\)
\(42\) 0 0
\(43\) −7.71890 − 5.40483i −1.17712 − 0.824230i −0.189623 − 0.981857i \(-0.560727\pi\)
−0.987499 + 0.157627i \(0.949615\pi\)
\(44\) 0 0
\(45\) 1.77930 + 3.80786i 0.265243 + 0.567642i
\(46\) 0 0
\(47\) 0.378221 + 4.32309i 0.0551693 + 0.630588i 0.972568 + 0.232618i \(0.0747293\pi\)
−0.917399 + 0.397969i \(0.869715\pi\)
\(48\) 0 0
\(49\) −5.52082 + 3.18745i −0.788689 + 0.455350i
\(50\) 0 0
\(51\) −2.26182 − 6.21429i −0.316718 − 0.870175i
\(52\) 0 0
\(53\) −0.408759 + 0.286216i −0.0561474 + 0.0393148i −0.601313 − 0.799013i \(-0.705356\pi\)
0.545166 + 0.838328i \(0.316467\pi\)
\(54\) 0 0
\(55\) −9.54433 + 0.842633i −1.28696 + 0.113621i
\(56\) 0 0
\(57\) 4.45787 − 1.18902i 0.590459 − 0.157489i
\(58\) 0 0
\(59\) 9.10551 − 7.64043i 1.18544 − 0.994699i 0.185509 − 0.982643i \(-0.440607\pi\)
0.999927 − 0.0120561i \(-0.00383767\pi\)
\(60\) 0 0
\(61\) −1.00870 + 5.72060i −0.129150 + 0.732448i 0.849606 + 0.527418i \(0.176840\pi\)
−0.978756 + 0.205029i \(0.934271\pi\)
\(62\) 0 0
\(63\) −0.628064 + 1.34689i −0.0791287 + 0.169692i
\(64\) 0 0
\(65\) −4.93214 + 4.14522i −0.611757 + 0.514151i
\(66\) 0 0
\(67\) 4.30306 − 0.376469i 0.525702 − 0.0459930i 0.178784 − 0.983888i \(-0.442784\pi\)
0.346918 + 0.937895i \(0.387228\pi\)
\(68\) 0 0
\(69\) −1.73019 + 2.99678i −0.208290 + 0.360769i
\(70\) 0 0
\(71\) −15.2125 + 2.68238i −1.80540 + 0.318340i −0.972114 − 0.234509i \(-0.924652\pi\)
−0.833282 + 0.552849i \(0.813541\pi\)
\(72\) 0 0
\(73\) −12.8258 + 5.98076i −1.50114 + 0.699995i −0.987443 − 0.157974i \(-0.949504\pi\)
−0.513701 + 0.857969i \(0.671726\pi\)
\(74\) 0 0
\(75\) −2.22903 + 4.79998i −0.257386 + 0.554254i
\(76\) 0 0
\(77\) −2.39556 − 2.39556i −0.272999 − 0.272999i
\(78\) 0 0
\(79\) −1.72702 − 0.628586i −0.194305 − 0.0707214i 0.243035 − 0.970018i \(-0.421857\pi\)
−0.437340 + 0.899296i \(0.644079\pi\)
\(80\) 0 0
\(81\) −0.0298904 − 0.169517i −0.00332115 − 0.0188352i
\(82\) 0 0
\(83\) −2.44614 − 9.12912i −0.268499 − 1.00205i −0.960074 − 0.279746i \(-0.909750\pi\)
0.691575 − 0.722304i \(-0.256917\pi\)
\(84\) 0 0
\(85\) −6.97575 + 12.1045i −0.756626 + 1.31291i
\(86\) 0 0
\(87\) 2.28172 − 8.51548i 0.244626 − 0.912955i
\(88\) 0 0
\(89\) 12.0968 − 4.40288i 1.28226 − 0.466705i 0.391081 − 0.920356i \(-0.372101\pi\)
0.891179 + 0.453652i \(0.149879\pi\)
\(90\) 0 0
\(91\) −2.24343 − 0.395577i −0.235175 − 0.0414677i
\(92\) 0 0
\(93\) 0.451282 − 5.15817i 0.0467957 − 0.534878i
\(94\) 0 0
\(95\) −7.98213 − 5.59335i −0.818950 − 0.573865i
\(96\) 0 0
\(97\) 1.34195 − 15.3386i 0.136254 − 1.55740i −0.553189 − 0.833056i \(-0.686589\pi\)
0.689443 − 0.724340i \(-0.257855\pi\)
\(98\) 0 0
\(99\) 7.93192 + 1.39861i 0.797188 + 0.140566i
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.317.4 yes 120
5.3 odd 4 inner 380.2.bh.a.13.7 ✓ 120
19.3 odd 18 inner 380.2.bh.a.117.7 yes 120
95.3 even 36 inner 380.2.bh.a.193.4 yes 120
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
380.2.bh.a.13.7 ✓ 120 5.3 odd 4 inner
380.2.bh.a.117.7 yes 120 19.3 odd 18 inner
380.2.bh.a.193.4 yes 120 95.3 even 36 inner
380.2.bh.a.317.4 yes 120 1.1 even 1 trivial