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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 13.7
Character \(\chi\) \(=\) 380.13
Dual form 380.2.bh.a.117.7

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.959290 - 0.447324i) q^{3} +(1.57989 + 1.58239i) q^{5} +(0.204631 + 0.763694i) q^{7} +(-1.20823 + 1.43991i) q^{9} +(2.14248 + 3.71088i) q^{11} +(-1.21768 + 2.61133i) q^{13} +(2.22341 + 0.811249i) q^{15} +(0.544538 - 6.22409i) q^{17} +(0.761826 - 4.29181i) q^{19} +(0.537919 + 0.641067i) q^{21} +(2.67802 - 1.87517i) q^{23} +(-0.00791432 + 4.99999i) q^{25} +(-1.33678 + 4.98893i) q^{27} +(-6.38036 - 5.35376i) q^{29} +(4.23651 + 2.44595i) q^{31} +(3.71522 + 2.60143i) q^{33} +(-0.885167 + 1.53036i) q^{35} +(-1.75916 - 1.75916i) q^{37} +3.04972i q^{39} +(0.896173 - 2.46221i) q^{41} +(5.40483 - 7.71890i) q^{43} +(-4.18735 + 0.363007i) q^{45} +(4.32309 - 0.378221i) q^{47} +(5.52082 - 3.18745i) q^{49} +(-2.26182 - 6.21429i) q^{51} +(-0.286216 - 0.408759i) q^{53} +(-2.48719 + 9.25301i) q^{55} +(-1.18902 - 4.45787i) q^{57} +(-9.10551 + 7.64043i) q^{59} +(-1.00870 + 5.72060i) q^{61} +(-1.34689 - 0.628064i) q^{63} +(-6.05594 + 2.19875i) q^{65} +(-0.376469 - 4.30306i) q^{67} +(1.73019 - 2.99678i) q^{69} +(-15.2125 + 2.68238i) q^{71} +(-5.98076 - 12.8258i) q^{73} +(2.22903 + 4.79998i) q^{75} +(-2.39556 + 2.39556i) q^{77} +(1.72702 + 0.628586i) q^{79} +(-0.0298904 - 0.169517i) q^{81} +(9.12912 - 2.44614i) q^{83} +(10.7092 - 8.97169i) q^{85} +(-8.51548 - 2.28172i) q^{87} +(-12.0968 + 4.40288i) q^{89} +(-2.24343 - 0.395577i) q^{91} +(5.15817 + 0.451282i) q^{93} +(7.99491 - 5.57507i) q^{95} +(-15.3386 - 1.34195i) q^{97} +(-7.93192 - 1.39861i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 120 q + 6 q^{7} + 18 q^{15} - 18 q^{17} + 48 q^{21} - 36 q^{23} - 24 q^{25} - 60 q^{33} - 18 q^{35} - 12 q^{41} - 36 q^{43} + 18 q^{45} - 24 q^{47} + 96 q^{51} - 18 q^{53} + 72 q^{55} - 6 q^{57} - 24 q^{61}+ \cdots - 60 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(21\) \(77\) \(191\)
\(\chi(n)\) \(e\left(\frac{5}{18}\right)\) \(e\left(\frac{3}{4}\right)\) \(1\)

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
\(3\) 0.959290 − 0.447324i 0.553846 − 0.258263i −0.125484 − 0.992096i \(-0.540048\pi\)
0.679330 + 0.733833i \(0.262271\pi\)
\(4\) 0 0
\(5\) 1.57989 + 1.58239i 0.706547 + 0.707666i
\(6\) 0 0
\(7\) 0.204631 + 0.763694i 0.0773433 + 0.288649i 0.993754 − 0.111589i \(-0.0355940\pi\)
−0.916411 + 0.400238i \(0.868927\pi\)
\(8\) 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
\(13\) −1.21768 + 2.61133i −0.337724 + 0.724252i −0.999680 − 0.0252910i \(-0.991949\pi\)
0.661956 + 0.749543i \(0.269727\pi\)
\(14\) 0 0
\(15\) 2.22341 + 0.811249i 0.574082 + 0.209463i
\(16\) 0 0
\(17\) 0.544538 − 6.22409i 0.132070 − 1.50956i −0.584125 − 0.811664i \(-0.698562\pi\)
0.716195 − 0.697900i \(-0.245882\pi\)
\(18\) 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
\(23\) 2.67802 − 1.87517i 0.558406 − 0.391000i −0.260028 − 0.965601i \(-0.583732\pi\)
0.818434 + 0.574601i \(0.194843\pi\)
\(24\) 0 0
\(25\) −0.00791432 + 4.99999i −0.00158286 + 0.999999i
\(26\) 0 0
\(27\) −1.33678 + 4.98893i −0.257264 + 0.960121i
\(28\) 0 0
\(29\) −6.38036 − 5.35376i −1.18480 − 0.994168i −0.999935 − 0.0114049i \(-0.996370\pi\)
−0.184869 − 0.982763i \(-0.559186\pi\)
\(30\) 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
\(33\) 3.71522 + 2.60143i 0.646738 + 0.452851i
\(34\) 0 0
\(35\) −0.885167 + 1.53036i −0.149621 + 0.258677i
\(36\) 0 0
\(37\) −1.75916 − 1.75916i −0.289204 − 0.289204i 0.547561 − 0.836766i \(-0.315556\pi\)
−0.836766 + 0.547561i \(0.815556\pi\)
\(38\) 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
\(43\) 5.40483 − 7.71890i 0.824230 − 1.17712i −0.157627 − 0.987499i \(-0.550385\pi\)
0.981857 − 0.189623i \(-0.0607266\pi\)
\(44\) 0 0
\(45\) −4.18735 + 0.363007i −0.624214 + 0.0541138i
\(46\) 0 0
\(47\) 4.32309 − 0.378221i 0.630588 − 0.0551693i 0.232618 − 0.972568i \(-0.425271\pi\)
0.397969 + 0.917399i \(0.369715\pi\)
\(48\) 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
\(53\) −0.286216 − 0.408759i −0.0393148 − 0.0561474i 0.799013 − 0.601313i \(-0.205356\pi\)
−0.838328 + 0.545166i \(0.816467\pi\)
\(54\) 0 0
\(55\) −2.48719 + 9.25301i −0.335372 + 1.24768i
\(56\) 0 0
\(57\) −1.18902 − 4.45787i −0.157489 − 0.590459i
\(58\) 0 0
\(59\) −9.10551 + 7.64043i −1.18544 + 0.994699i −0.185509 + 0.982643i \(0.559393\pi\)
−0.999927 + 0.0120561i \(0.996162\pi\)
\(60\) 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
\(63\) −1.34689 − 0.628064i −0.169692 − 0.0791287i
\(64\) 0 0
\(65\) −6.05594 + 2.19875i −0.751146 + 0.272722i
\(66\) 0 0
\(67\) −0.376469 − 4.30306i −0.0459930 − 0.525702i −0.983888 − 0.178784i \(-0.942784\pi\)
0.937895 − 0.346918i \(-0.112772\pi\)
\(68\) 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
\(73\) −5.98076 − 12.8258i −0.699995 − 1.50114i −0.857969 − 0.513701i \(-0.828274\pi\)
0.157974 − 0.987443i \(-0.449504\pi\)
\(74\) 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 0.437340 − 0.899296i \(-0.355921\pi\)
−0.243035 + 0.970018i \(0.578143\pi\)
\(80\) 0 0
\(81\) −0.0298904 − 0.169517i −0.00332115 − 0.0188352i
\(82\) 0 0
\(83\) 9.12912 − 2.44614i 1.00205 − 0.268499i 0.279746 − 0.960074i \(-0.409750\pi\)
0.722304 + 0.691575i \(0.243083\pi\)
\(84\) 0 0
\(85\) 10.7092 − 8.97169i 1.16158 − 0.973117i
\(86\) 0 0
\(87\) −8.51548 − 2.28172i −0.912955 − 0.244626i
\(88\) 0 0
\(89\) −12.0968 + 4.40288i −1.28226 + 0.466705i −0.891179 − 0.453652i \(-0.850121\pi\)
−0.391081 + 0.920356i \(0.627899\pi\)
\(90\) 0 0
\(91\) −2.24343 − 0.395577i −0.235175 − 0.0414677i
\(92\) 0 0
\(93\) 5.15817 + 0.451282i 0.534878 + 0.0467957i
\(94\) 0 0
\(95\) 7.99491 − 5.57507i 0.820261 − 0.571990i
\(96\) 0 0
\(97\) −15.3386 − 1.34195i −1.55740 − 0.136254i −0.724340 − 0.689443i \(-0.757855\pi\)
−0.833056 + 0.553189i \(0.813411\pi\)
\(98\) 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.13.7 ✓ 120
5.2 odd 4 inner 380.2.bh.a.317.4 yes 120
19.3 odd 18 inner 380.2.bh.a.193.4 yes 120
95.22 even 36 inner 380.2.bh.a.117.7 yes 120
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
380.2.bh.a.13.7 ✓ 120 1.1 even 1 trivial
380.2.bh.a.117.7 yes 120 95.22 even 36 inner
380.2.bh.a.193.4 yes 120 19.3 odd 18 inner
380.2.bh.a.317.4 yes 120 5.2 odd 4 inner