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(267,380)] 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("380.267"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(380, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([2, 1, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 380 = 2^{2} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 380.k (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(3.03431527681\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 343.1
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 380.343
Dual form 380.2.k.a.267.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-1.00000 - 1.00000i) q^{2} +(-1.00000 + 1.00000i) q^{3} +2.00000i q^{4} +(-2.00000 + 1.00000i) q^{5} +2.00000 q^{6} +(-2.00000 - 2.00000i) q^{7} +(2.00000 - 2.00000i) q^{8} +1.00000i q^{9} +(3.00000 + 1.00000i) q^{10} +(-2.00000 - 2.00000i) q^{12} +4.00000i q^{14} +(1.00000 - 3.00000i) q^{15} -4.00000 q^{16} +(5.00000 - 5.00000i) q^{17} +(1.00000 - 1.00000i) q^{18} -1.00000 q^{19} +(-2.00000 - 4.00000i) q^{20} +4.00000 q^{21} +(4.00000 - 4.00000i) q^{23} +4.00000i q^{24} +(3.00000 - 4.00000i) q^{25} +(-4.00000 - 4.00000i) q^{27} +(4.00000 - 4.00000i) q^{28} -6.00000i q^{29} +(-4.00000 + 2.00000i) q^{30} +(4.00000 + 4.00000i) q^{32} -10.0000 q^{34} +(6.00000 + 2.00000i) q^{35} -2.00000 q^{36} +(1.00000 + 1.00000i) q^{38} +(-2.00000 + 6.00000i) q^{40} +2.00000 q^{41} +(-4.00000 - 4.00000i) q^{42} +(-6.00000 + 6.00000i) q^{43} +(-1.00000 - 2.00000i) q^{45} -8.00000 q^{46} +(-2.00000 - 2.00000i) q^{47} +(4.00000 - 4.00000i) q^{48} +1.00000i q^{49} +(-7.00000 + 1.00000i) q^{50} +10.0000i q^{51} +(-10.0000 - 10.0000i) q^{53} +8.00000i q^{54} -8.00000 q^{56} +(1.00000 - 1.00000i) q^{57} +(-6.00000 + 6.00000i) q^{58} +10.0000 q^{59} +(6.00000 + 2.00000i) q^{60} +2.00000 q^{61} +(2.00000 - 2.00000i) q^{63} -8.00000i q^{64} +(3.00000 + 3.00000i) q^{67} +(10.0000 + 10.0000i) q^{68} +8.00000i q^{69} +(-4.00000 - 8.00000i) q^{70} +(2.00000 + 2.00000i) q^{72} +(5.00000 + 5.00000i) q^{73} +(1.00000 + 7.00000i) q^{75} -2.00000i q^{76} -10.0000 q^{79} +(8.00000 - 4.00000i) q^{80} +5.00000 q^{81} +(-2.00000 - 2.00000i) q^{82} +(4.00000 - 4.00000i) q^{83} +8.00000i q^{84} +(-5.00000 + 15.0000i) q^{85} +12.0000 q^{86} +(6.00000 + 6.00000i) q^{87} -6.00000i q^{89} +(-1.00000 + 3.00000i) q^{90} +(8.00000 + 8.00000i) q^{92} +4.00000i q^{94} +(2.00000 - 1.00000i) q^{95} -8.00000 q^{96} +(-10.0000 + 10.0000i) q^{97} +(1.00000 - 1.00000i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} - 2 q^{3} - 4 q^{5} + 4 q^{6} - 4 q^{7} + 4 q^{8} + 6 q^{10} - 4 q^{12} + 2 q^{15} - 8 q^{16} + 10 q^{17} + 2 q^{18} - 2 q^{19} - 4 q^{20} + 8 q^{21} + 8 q^{23} + 6 q^{25} - 8 q^{27} + 8 q^{28}+ \cdots + 2 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 − 1.00000i −0.707107 − 0.707107i
\(3\) −1.00000 + 1.00000i −0.577350 + 0.577350i −0.934172 − 0.356822i \(-0.883860\pi\)
0.356822 + 0.934172i \(0.383860\pi\)
\(4\) 2.00000i 1.00000i
\(5\) −2.00000 + 1.00000i −0.894427 + 0.447214i
\(6\) 2.00000 0.816497
\(7\) −2.00000 − 2.00000i −0.755929 − 0.755929i 0.219650 − 0.975579i \(-0.429509\pi\)
−0.975579 + 0.219650i \(0.929509\pi\)
\(8\) 2.00000 − 2.00000i 0.707107 − 0.707107i
\(9\) 1.00000i 0.333333i
\(10\) 3.00000 + 1.00000i 0.948683 + 0.316228i
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) −2.00000 − 2.00000i −0.577350 − 0.577350i
\(13\) 0 0 0.707107 − 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(14\) 4.00000i 1.06904i
\(15\) 1.00000 − 3.00000i 0.258199 − 0.774597i
\(16\) −4.00000 −1.00000
\(17\) 5.00000 − 5.00000i 1.21268 − 1.21268i 0.242536 − 0.970143i \(-0.422021\pi\)
0.970143 − 0.242536i \(-0.0779791\pi\)
\(18\) 1.00000 − 1.00000i 0.235702 − 0.235702i
\(19\) −1.00000 −0.229416
\(20\) −2.00000 − 4.00000i −0.447214 − 0.894427i
\(21\) 4.00000 0.872872
\(22\) 0 0
\(23\) 4.00000 − 4.00000i 0.834058 − 0.834058i −0.154011 − 0.988069i \(-0.549219\pi\)
0.988069 + 0.154011i \(0.0492193\pi\)
\(24\) 4.00000i 0.816497i
\(25\) 3.00000 − 4.00000i 0.600000 − 0.800000i
\(26\) 0 0
\(27\) −4.00000 − 4.00000i −0.769800 − 0.769800i
\(28\) 4.00000 − 4.00000i 0.755929 − 0.755929i
\(29\) − 6.00000i − 1.11417i −0.830455 − 0.557086i \(-0.811919\pi\)
0.830455 − 0.557086i \(-0.188081\pi\)
\(30\) −4.00000 + 2.00000i −0.730297 + 0.365148i
\(31\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(32\) 4.00000 + 4.00000i 0.707107 + 0.707107i
\(33\) 0 0
\(34\) −10.0000 −1.71499
\(35\) 6.00000 + 2.00000i 1.01419 + 0.338062i
\(36\) −2.00000 −0.333333
\(37\) 0 0 −0.707107 − 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(38\) 1.00000 + 1.00000i 0.162221 + 0.162221i
\(39\) 0 0
\(40\) −2.00000 + 6.00000i −0.316228 + 0.948683i
\(41\) 2.00000 0.312348 0.156174 − 0.987730i \(-0.450084\pi\)
0.156174 + 0.987730i \(0.450084\pi\)
\(42\) −4.00000 − 4.00000i −0.617213 − 0.617213i
\(43\) −6.00000 + 6.00000i −0.914991 + 0.914991i −0.996660 − 0.0816682i \(-0.973975\pi\)
0.0816682 + 0.996660i \(0.473975\pi\)
\(44\) 0 0
\(45\) −1.00000 − 2.00000i −0.149071 − 0.298142i
\(46\) −8.00000 −1.17954
\(47\) −2.00000 − 2.00000i −0.291730 − 0.291730i 0.546033 − 0.837763i \(-0.316137\pi\)
−0.837763 + 0.546033i \(0.816137\pi\)
\(48\) 4.00000 − 4.00000i 0.577350 − 0.577350i
\(49\) 1.00000i 0.142857i
\(50\) −7.00000 + 1.00000i −0.989949 + 0.141421i
\(51\) 10.0000i 1.40028i
\(52\) 0 0
\(53\) −10.0000 − 10.0000i −1.37361 − 1.37361i −0.855034 − 0.518571i \(-0.826464\pi\)
−0.518571 − 0.855034i \(-0.673536\pi\)
\(54\) 8.00000i 1.08866i
\(55\) 0 0
\(56\) −8.00000 −1.06904
\(57\) 1.00000 − 1.00000i 0.132453 − 0.132453i
\(58\) −6.00000 + 6.00000i −0.787839 + 0.787839i
\(59\) 10.0000 1.30189 0.650945 − 0.759125i \(-0.274373\pi\)
0.650945 + 0.759125i \(0.274373\pi\)
\(60\) 6.00000 + 2.00000i 0.774597 + 0.258199i
\(61\) 2.00000 0.256074 0.128037 − 0.991769i \(-0.459132\pi\)
0.128037 + 0.991769i \(0.459132\pi\)
\(62\) 0 0
\(63\) 2.00000 − 2.00000i 0.251976 − 0.251976i
\(64\) − 8.00000i − 1.00000i
\(65\) 0 0
\(66\) 0 0
\(67\) 3.00000 + 3.00000i 0.366508 + 0.366508i 0.866202 − 0.499694i \(-0.166554\pi\)
−0.499694 + 0.866202i \(0.666554\pi\)
\(68\) 10.0000 + 10.0000i 1.21268 + 1.21268i
\(69\) 8.00000i 0.963087i
\(70\) −4.00000 − 8.00000i −0.478091 − 0.956183i
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 2.00000 + 2.00000i 0.235702 + 0.235702i
\(73\) 5.00000 + 5.00000i 0.585206 + 0.585206i 0.936329 − 0.351123i \(-0.114200\pi\)
−0.351123 + 0.936329i \(0.614200\pi\)
\(74\) 0 0
\(75\) 1.00000 + 7.00000i 0.115470 + 0.808290i
\(76\) − 2.00000i − 0.229416i
\(77\) 0 0
\(78\) 0 0
\(79\) −10.0000 −1.12509 −0.562544 − 0.826767i \(-0.690177\pi\)
−0.562544 + 0.826767i \(0.690177\pi\)
\(80\) 8.00000 − 4.00000i 0.894427 − 0.447214i
\(81\) 5.00000 0.555556
\(82\) −2.00000 − 2.00000i −0.220863 − 0.220863i
\(83\) 4.00000 − 4.00000i 0.439057 − 0.439057i −0.452638 − 0.891695i \(-0.649517\pi\)
0.891695 + 0.452638i \(0.149517\pi\)
\(84\) 8.00000i 0.872872i
\(85\) −5.00000 + 15.0000i −0.542326 + 1.62698i
\(86\) 12.0000 1.29399
\(87\) 6.00000 + 6.00000i 0.643268 + 0.643268i
\(88\) 0 0
\(89\) − 6.00000i − 0.635999i −0.948091 − 0.317999i \(-0.896989\pi\)
0.948091 − 0.317999i \(-0.103011\pi\)
\(90\) −1.00000 + 3.00000i −0.105409 + 0.316228i
\(91\) 0 0
\(92\) 8.00000 + 8.00000i 0.834058 + 0.834058i
\(93\) 0 0
\(94\) 4.00000i 0.412568i
\(95\) 2.00000 − 1.00000i 0.205196 − 0.102598i
\(96\) −8.00000 −0.816497
\(97\) −10.0000 + 10.0000i −1.01535 + 1.01535i −0.0154658 + 0.999880i \(0.504923\pi\)
−0.999880 + 0.0154658i \(0.995077\pi\)
\(98\) 1.00000 − 1.00000i 0.101015 − 0.101015i
\(99\) 0 0
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.k.a.343.1 yes 2
4.3 odd 2 380.2.k.b.343.1 yes 2
5.2 odd 4 380.2.k.b.267.1 yes 2
20.7 even 4 inner 380.2.k.a.267.1 ✓ 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
380.2.k.a.267.1 ✓ 2 20.7 even 4 inner
380.2.k.a.343.1 yes 2 1.1 even 1 trivial
380.2.k.b.267.1 yes 2 5.2 odd 4
380.2.k.b.343.1 yes 2 4.3 odd 2