Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [380,2,Mod(49,380)] 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("380.49"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(380, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 3, 4])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 380 = 2^{2} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 380.r (of order \(6\), degree \(2\), 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: \(20\)
Relative dimension: \(10\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - 20 x^{18} + 261 x^{16} - 1994 x^{14} + 11074 x^{12} - 39211 x^{10} + 99376 x^{8} - 134299 x^{6} + \cdots + 4096 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 349.6
Root \(0.392182 + 0.226426i\) of defining polynomial
Character \(\chi\) \(=\) 380.349
Dual form 380.2.r.a.49.6

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.392182 - 0.226426i) q^{3} +(0.207009 + 2.22647i) q^{5} -2.54366i q^{7} +(-1.39746 + 2.42048i) q^{9} +2.22377 q^{11} +(6.08116 + 3.51096i) q^{13} +(0.585315 + 0.826307i) q^{15} +(2.21492 - 1.27878i) q^{17} +(2.70498 + 3.41805i) q^{19} +(-0.575952 - 0.997578i) q^{21} +(-6.95328 - 4.01448i) q^{23} +(-4.91429 + 0.921799i) q^{25} +2.62425i q^{27} +(-0.941734 + 1.63113i) q^{29} +5.98111 q^{31} +(0.872121 - 0.503519i) q^{33} +(5.66338 - 0.526562i) q^{35} +2.86105i q^{37} +3.17989 q^{39} +(-3.67524 - 6.36571i) q^{41} +(3.19919 - 1.84706i) q^{43} +(-5.67839 - 2.61034i) q^{45} +(4.09540 + 2.36448i) q^{47} +0.529782 q^{49} +(0.579100 - 1.00303i) q^{51} +(-8.91226 - 5.14549i) q^{53} +(0.460341 + 4.95114i) q^{55} +(1.83478 + 0.728020i) q^{57} +(-3.73666 - 6.47208i) q^{59} +(4.17839 - 7.23719i) q^{61} +(6.15687 + 3.55467i) q^{63} +(-6.55817 + 14.2663i) q^{65} +(-10.7040 - 6.17997i) q^{67} -3.63593 q^{69} +(-4.13931 - 7.16950i) q^{71} +(-10.9489 + 6.32134i) q^{73} +(-1.71858 + 1.47424i) q^{75} -5.65651i q^{77} +(-2.13067 - 3.69043i) q^{79} +(-3.59819 - 6.23225i) q^{81} +14.7613i q^{83} +(3.30568 + 4.66672i) q^{85} +0.852933i q^{87} +(-7.19403 + 12.4604i) q^{89} +(8.93069 - 15.4684i) q^{91} +(2.34568 - 1.35428i) q^{93} +(-7.05022 + 6.73011i) q^{95} +(5.04871 - 2.91488i) q^{97} +(-3.10763 + 5.38258i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + q^{5} + 10 q^{9} - 5 q^{15} + 14 q^{19} - 8 q^{21} + 9 q^{25} - 16 q^{29} + 8 q^{31} - 2 q^{35} - 8 q^{39} + 26 q^{41} - 32 q^{45} - 44 q^{49} + 26 q^{51} - 12 q^{55} + 4 q^{59} + 2 q^{61} - 18 q^{65}+ \cdots - 20 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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
\(3\) 0.392182 − 0.226426i 0.226426 − 0.130727i −0.382496 − 0.923957i \(-0.624935\pi\)
0.608922 + 0.793230i \(0.291602\pi\)
\(4\) 0 0
\(5\) 0.207009 + 2.22647i 0.0925774 + 0.995705i
\(6\) 0 0
\(7\) − 2.54366i − 0.961414i −0.876881 − 0.480707i \(-0.840380\pi\)
0.876881 − 0.480707i \(-0.159620\pi\)
\(8\) 0 0
\(9\) −1.39746 + 2.42048i −0.465821 + 0.806825i
\(10\) 0 0
\(11\) 2.22377 0.670491 0.335246 − 0.942131i \(-0.391181\pi\)
0.335246 + 0.942131i \(0.391181\pi\)
\(12\) 0 0
\(13\) 6.08116 + 3.51096i 1.68661 + 0.973765i 0.957084 + 0.289812i \(0.0935927\pi\)
0.729526 + 0.683953i \(0.239741\pi\)
\(14\) 0 0
\(15\) 0.585315 + 0.826307i 0.151128 + 0.213351i
\(16\) 0 0
\(17\) 2.21492 − 1.27878i 0.537197 − 0.310151i −0.206745 − 0.978395i \(-0.566287\pi\)
0.743942 + 0.668244i \(0.232954\pi\)
\(18\) 0 0
\(19\) 2.70498 + 3.41805i 0.620565 + 0.784155i
\(20\) 0 0
\(21\) −0.575952 − 0.997578i −0.125683 − 0.217689i
\(22\) 0 0
\(23\) −6.95328 − 4.01448i −1.44986 − 0.837076i −0.451387 − 0.892329i \(-0.649070\pi\)
−0.998472 + 0.0552521i \(0.982404\pi\)
\(24\) 0 0
\(25\) −4.91429 + 0.921799i −0.982859 + 0.184360i
\(26\) 0 0
\(27\) 2.62425i 0.505036i
\(28\) 0 0
\(29\) −0.941734 + 1.63113i −0.174876 + 0.302893i −0.940118 − 0.340849i \(-0.889286\pi\)
0.765243 + 0.643742i \(0.222619\pi\)
\(30\) 0 0
\(31\) 5.98111 1.07424 0.537120 − 0.843506i \(-0.319512\pi\)
0.537120 + 0.843506i \(0.319512\pi\)
\(32\) 0 0
\(33\) 0.872121 − 0.503519i 0.151817 − 0.0876515i
\(34\) 0 0
\(35\) 5.66338 − 0.526562i 0.957285 − 0.0890052i
\(36\) 0 0
\(37\) 2.86105i 0.470353i 0.971953 + 0.235177i \(0.0755669\pi\)
−0.971953 + 0.235177i \(0.924433\pi\)
\(38\) 0 0
\(39\) 3.17989 0.509190
\(40\) 0 0
\(41\) −3.67524 − 6.36571i −0.573977 − 0.994157i −0.996152 − 0.0876426i \(-0.972067\pi\)
0.422175 − 0.906514i \(-0.361267\pi\)
\(42\) 0 0
\(43\) 3.19919 − 1.84706i 0.487873 − 0.281673i −0.235819 − 0.971797i \(-0.575777\pi\)
0.723691 + 0.690124i \(0.242444\pi\)
\(44\) 0 0
\(45\) −5.67839 − 2.61034i −0.846485 − 0.389126i
\(46\) 0 0
\(47\) 4.09540 + 2.36448i 0.597376 + 0.344895i 0.768008 − 0.640440i \(-0.221248\pi\)
−0.170633 + 0.985335i \(0.554581\pi\)
\(48\) 0 0
\(49\) 0.529782 0.0756832
\(50\) 0 0
\(51\) 0.579100 − 1.00303i 0.0810903 − 0.140453i
\(52\) 0 0
\(53\) −8.91226 − 5.14549i −1.22419 − 0.706788i −0.258383 − 0.966042i \(-0.583190\pi\)
−0.965809 + 0.259255i \(0.916523\pi\)
\(54\) 0 0
\(55\) 0.460341 + 4.95114i 0.0620724 + 0.667612i
\(56\) 0 0
\(57\) 1.83478 + 0.728020i 0.243023 + 0.0964286i
\(58\) 0 0
\(59\) −3.73666 − 6.47208i −0.486472 − 0.842593i 0.513408 − 0.858145i \(-0.328383\pi\)
−0.999879 + 0.0155515i \(0.995050\pi\)
\(60\) 0 0
\(61\) 4.17839 − 7.23719i 0.534988 − 0.926627i −0.464176 − 0.885743i \(-0.653649\pi\)
0.999164 − 0.0408838i \(-0.0130174\pi\)
\(62\) 0 0
\(63\) 6.15687 + 3.55467i 0.775693 + 0.447847i
\(64\) 0 0
\(65\) −6.55817 + 14.2663i −0.813441 + 1.76952i
\(66\) 0 0
\(67\) −10.7040 − 6.17997i −1.30771 − 0.755004i −0.325993 − 0.945372i \(-0.605698\pi\)
−0.981713 + 0.190368i \(0.939032\pi\)
\(68\) 0 0
\(69\) −3.63593 −0.437715
\(70\) 0 0
\(71\) −4.13931 − 7.16950i −0.491246 − 0.850863i 0.508703 − 0.860942i \(-0.330125\pi\)
−0.999949 + 0.0100790i \(0.996792\pi\)
\(72\) 0 0
\(73\) −10.9489 + 6.32134i −1.28147 + 0.739857i −0.977117 − 0.212703i \(-0.931773\pi\)
−0.304352 + 0.952559i \(0.598440\pi\)
\(74\) 0 0
\(75\) −1.71858 + 1.47424i −0.198444 + 0.170230i
\(76\) 0 0
\(77\) − 5.65651i − 0.644620i
\(78\) 0 0
\(79\) −2.13067 − 3.69043i −0.239719 − 0.415206i 0.720914 − 0.693024i \(-0.243722\pi\)
−0.960634 + 0.277818i \(0.910389\pi\)
\(80\) 0 0
\(81\) −3.59819 − 6.23225i −0.399799 − 0.692472i
\(82\) 0 0
\(83\) 14.7613i 1.62026i 0.586248 + 0.810132i \(0.300604\pi\)
−0.586248 + 0.810132i \(0.699396\pi\)
\(84\) 0 0
\(85\) 3.30568 + 4.66672i 0.358551 + 0.506177i
\(86\) 0 0
\(87\) 0.852933i 0.0914440i
\(88\) 0 0
\(89\) −7.19403 + 12.4604i −0.762566 + 1.32080i 0.178958 + 0.983857i \(0.442727\pi\)
−0.941524 + 0.336946i \(0.890606\pi\)
\(90\) 0 0
\(91\) 8.93069 − 15.4684i 0.936191 − 1.62153i
\(92\) 0 0
\(93\) 2.34568 − 1.35428i 0.243236 − 0.140432i
\(94\) 0 0
\(95\) −7.05022 + 6.73011i −0.723337 + 0.690495i
\(96\) 0 0
\(97\) 5.04871 − 2.91488i 0.512619 − 0.295961i −0.221291 − 0.975208i \(-0.571027\pi\)
0.733910 + 0.679247i \(0.237694\pi\)
\(98\) 0 0
\(99\) −3.10763 + 5.38258i −0.312329 + 0.540969i
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.r.a.349.6 yes 20
3.2 odd 2 3420.2.bj.c.2629.5 20
5.2 odd 4 1900.2.i.g.501.5 20
5.3 odd 4 1900.2.i.g.501.6 20
5.4 even 2 inner 380.2.r.a.349.5 yes 20
15.14 odd 2 3420.2.bj.c.2629.3 20
19.11 even 3 inner 380.2.r.a.49.5 ✓ 20
57.11 odd 6 3420.2.bj.c.1189.3 20
95.49 even 6 inner 380.2.r.a.49.6 yes 20
95.68 odd 12 1900.2.i.g.201.6 20
95.87 odd 12 1900.2.i.g.201.5 20
285.239 odd 6 3420.2.bj.c.1189.5 20
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
380.2.r.a.49.5 ✓ 20 19.11 even 3 inner
380.2.r.a.49.6 yes 20 95.49 even 6 inner
380.2.r.a.349.5 yes 20 5.4 even 2 inner
380.2.r.a.349.6 yes 20 1.1 even 1 trivial
1900.2.i.g.201.5 20 95.87 odd 12
1900.2.i.g.201.6 20 95.68 odd 12
1900.2.i.g.501.5 20 5.2 odd 4
1900.2.i.g.501.6 20 5.3 odd 4
3420.2.bj.c.1189.3 20 57.11 odd 6
3420.2.bj.c.1189.5 20 285.239 odd 6
3420.2.bj.c.2629.3 20 15.14 odd 2
3420.2.bj.c.2629.5 20 3.2 odd 2