Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1950,2,Mod(451,1950)] 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("1950.451"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1950, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 4])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1950 = 2 \cdot 3 \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1950.i (of order \(3\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,-1,1,-1,0,1,2,2,-1,0,-6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(15.5708283941\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 78)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 601.1
Root \(0.500000 - 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 1950.601
Dual form 1950.2.i.m.451.1

$q$-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+(-0.500000 - 0.866025i) q^{2} +(0.500000 + 0.866025i) q^{3} +(-0.500000 + 0.866025i) q^{4} +(0.500000 - 0.866025i) q^{6} +(1.00000 - 1.73205i) q^{7} +1.00000 q^{8} +(-0.500000 + 0.866025i) q^{9} +(-3.00000 - 5.19615i) q^{11} -1.00000 q^{12} +(3.50000 - 0.866025i) q^{13} -2.00000 q^{14} +(-0.500000 - 0.866025i) q^{16} +(-1.50000 + 2.59808i) q^{17} +1.00000 q^{18} +(-1.00000 + 1.73205i) q^{19} +2.00000 q^{21} +(-3.00000 + 5.19615i) q^{22} +(-3.00000 - 5.19615i) q^{23} +(0.500000 + 0.866025i) q^{24} +(-2.50000 - 2.59808i) q^{26} -1.00000 q^{27} +(1.00000 + 1.73205i) q^{28} +(-1.50000 - 2.59808i) q^{29} -4.00000 q^{31} +(-0.500000 + 0.866025i) q^{32} +(3.00000 - 5.19615i) q^{33} +3.00000 q^{34} +(-0.500000 - 0.866025i) q^{36} +(-3.50000 - 6.06218i) q^{37} +2.00000 q^{38} +(2.50000 + 2.59808i) q^{39} +(1.50000 + 2.59808i) q^{41} +(-1.00000 - 1.73205i) q^{42} +(-5.00000 + 8.66025i) q^{43} +6.00000 q^{44} +(-3.00000 + 5.19615i) q^{46} -6.00000 q^{47} +(0.500000 - 0.866025i) q^{48} +(1.50000 + 2.59808i) q^{49} -3.00000 q^{51} +(-1.00000 + 3.46410i) q^{52} -3.00000 q^{53} +(0.500000 + 0.866025i) q^{54} +(1.00000 - 1.73205i) q^{56} -2.00000 q^{57} +(-1.50000 + 2.59808i) q^{58} +(3.50000 - 6.06218i) q^{61} +(2.00000 + 3.46410i) q^{62} +(1.00000 + 1.73205i) q^{63} +1.00000 q^{64} -6.00000 q^{66} +(-5.00000 - 8.66025i) q^{67} +(-1.50000 - 2.59808i) q^{68} +(3.00000 - 5.19615i) q^{69} +(-3.00000 + 5.19615i) q^{71} +(-0.500000 + 0.866025i) q^{72} +13.0000 q^{73} +(-3.50000 + 6.06218i) q^{74} +(-1.00000 - 1.73205i) q^{76} -12.0000 q^{77} +(1.00000 - 3.46410i) q^{78} -4.00000 q^{79} +(-0.500000 - 0.866025i) q^{81} +(1.50000 - 2.59808i) q^{82} +6.00000 q^{83} +(-1.00000 + 1.73205i) q^{84} +10.0000 q^{86} +(1.50000 - 2.59808i) q^{87} +(-3.00000 - 5.19615i) q^{88} +(-9.00000 - 15.5885i) q^{89} +(2.00000 - 6.92820i) q^{91} +6.00000 q^{92} +(-2.00000 - 3.46410i) q^{93} +(3.00000 + 5.19615i) q^{94} -1.00000 q^{96} +(7.00000 - 12.1244i) q^{97} +(1.50000 - 2.59808i) q^{98} +6.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{2} + q^{3} - q^{4} + q^{6} + 2 q^{7} + 2 q^{8} - q^{9} - 6 q^{11} - 2 q^{12} + 7 q^{13} - 4 q^{14} - q^{16} - 3 q^{17} + 2 q^{18} - 2 q^{19} + 4 q^{21} - 6 q^{22} - 6 q^{23} + q^{24} - 5 q^{26}+ \cdots + 12 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1950\mathbb{Z}\right)^\times\).

\(n\) \(301\) \(1301\) \(1327\)
\(\chi(n)\) \(e\left(\frac{1}{3}\right)\) \(1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.500000 − 0.866025i −0.353553 − 0.612372i
\(3\) 0.500000 + 0.866025i 0.288675 + 0.500000i
\(4\) −0.500000 + 0.866025i −0.250000 + 0.433013i
\(5\) 0 0
\(6\) 0.500000 − 0.866025i 0.204124 − 0.353553i
\(7\) 1.00000 − 1.73205i 0.377964 − 0.654654i −0.612801 − 0.790237i \(-0.709957\pi\)
0.990766 + 0.135583i \(0.0432908\pi\)
\(8\) 1.00000 0.353553
\(9\) −0.500000 + 0.866025i −0.166667 + 0.288675i
\(10\) 0 0
\(11\) −3.00000 − 5.19615i −0.904534 − 1.56670i −0.821541 − 0.570149i \(-0.806886\pi\)
−0.0829925 − 0.996550i \(-0.526448\pi\)
\(12\) −1.00000 −0.288675
\(13\) 3.50000 − 0.866025i 0.970725 − 0.240192i
\(14\) −2.00000 −0.534522
\(15\) 0 0
\(16\) −0.500000 − 0.866025i −0.125000 − 0.216506i
\(17\) −1.50000 + 2.59808i −0.363803 + 0.630126i −0.988583 − 0.150675i \(-0.951855\pi\)
0.624780 + 0.780801i \(0.285189\pi\)
\(18\) 1.00000 0.235702
\(19\) −1.00000 + 1.73205i −0.229416 + 0.397360i −0.957635 − 0.287984i \(-0.907015\pi\)
0.728219 + 0.685344i \(0.240348\pi\)
\(20\) 0 0
\(21\) 2.00000 0.436436
\(22\) −3.00000 + 5.19615i −0.639602 + 1.10782i
\(23\) −3.00000 − 5.19615i −0.625543 − 1.08347i −0.988436 − 0.151642i \(-0.951544\pi\)
0.362892 − 0.931831i \(-0.381789\pi\)
\(24\) 0.500000 + 0.866025i 0.102062 + 0.176777i
\(25\) 0 0
\(26\) −2.50000 − 2.59808i −0.490290 − 0.509525i
\(27\) −1.00000 −0.192450
\(28\) 1.00000 + 1.73205i 0.188982 + 0.327327i
\(29\) −1.50000 − 2.59808i −0.278543 − 0.482451i 0.692480 − 0.721437i \(-0.256518\pi\)
−0.971023 + 0.238987i \(0.923185\pi\)
\(30\) 0 0
\(31\) −4.00000 −0.718421 −0.359211 − 0.933257i \(-0.616954\pi\)
−0.359211 + 0.933257i \(0.616954\pi\)
\(32\) −0.500000 + 0.866025i −0.0883883 + 0.153093i
\(33\) 3.00000 − 5.19615i 0.522233 − 0.904534i
\(34\) 3.00000 0.514496
\(35\) 0 0
\(36\) −0.500000 − 0.866025i −0.0833333 − 0.144338i
\(37\) −3.50000 − 6.06218i −0.575396 − 0.996616i −0.995998 − 0.0893706i \(-0.971514\pi\)
0.420602 − 0.907245i \(-0.361819\pi\)
\(38\) 2.00000 0.324443
\(39\) 2.50000 + 2.59808i 0.400320 + 0.416025i
\(40\) 0 0
\(41\) 1.50000 + 2.59808i 0.234261 + 0.405751i 0.959058 − 0.283211i \(-0.0913998\pi\)
−0.724797 + 0.688963i \(0.758066\pi\)
\(42\) −1.00000 − 1.73205i −0.154303 − 0.267261i
\(43\) −5.00000 + 8.66025i −0.762493 + 1.32068i 0.179069 + 0.983836i \(0.442691\pi\)
−0.941562 + 0.336840i \(0.890642\pi\)
\(44\) 6.00000 0.904534
\(45\) 0 0
\(46\) −3.00000 + 5.19615i −0.442326 + 0.766131i
\(47\) −6.00000 −0.875190 −0.437595 − 0.899172i \(-0.644170\pi\)
−0.437595 + 0.899172i \(0.644170\pi\)
\(48\) 0.500000 − 0.866025i 0.0721688 − 0.125000i
\(49\) 1.50000 + 2.59808i 0.214286 + 0.371154i
\(50\) 0 0
\(51\) −3.00000 −0.420084
\(52\) −1.00000 + 3.46410i −0.138675 + 0.480384i
\(53\) −3.00000 −0.412082 −0.206041 − 0.978543i \(-0.566058\pi\)
−0.206041 + 0.978543i \(0.566058\pi\)
\(54\) 0.500000 + 0.866025i 0.0680414 + 0.117851i
\(55\) 0 0
\(56\) 1.00000 − 1.73205i 0.133631 − 0.231455i
\(57\) −2.00000 −0.264906
\(58\) −1.50000 + 2.59808i −0.196960 + 0.341144i
\(59\) 0 0 −0.866025 − 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(60\) 0 0
\(61\) 3.50000 − 6.06218i 0.448129 − 0.776182i −0.550135 − 0.835076i \(-0.685424\pi\)
0.998264 + 0.0588933i \(0.0187572\pi\)
\(62\) 2.00000 + 3.46410i 0.254000 + 0.439941i
\(63\) 1.00000 + 1.73205i 0.125988 + 0.218218i
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) −6.00000 −0.738549
\(67\) −5.00000 − 8.66025i −0.610847 − 1.05802i −0.991098 − 0.133135i \(-0.957496\pi\)
0.380251 − 0.924883i \(-0.375838\pi\)
\(68\) −1.50000 − 2.59808i −0.181902 − 0.315063i
\(69\) 3.00000 − 5.19615i 0.361158 − 0.625543i
\(70\) 0 0
\(71\) −3.00000 + 5.19615i −0.356034 + 0.616670i −0.987294 − 0.158901i \(-0.949205\pi\)
0.631260 + 0.775571i \(0.282538\pi\)
\(72\) −0.500000 + 0.866025i −0.0589256 + 0.102062i
\(73\) 13.0000 1.52153 0.760767 − 0.649025i \(-0.224823\pi\)
0.760767 + 0.649025i \(0.224823\pi\)
\(74\) −3.50000 + 6.06218i −0.406867 + 0.704714i
\(75\) 0 0
\(76\) −1.00000 − 1.73205i −0.114708 − 0.198680i
\(77\) −12.0000 −1.36753
\(78\) 1.00000 − 3.46410i 0.113228 − 0.392232i
\(79\) −4.00000 −0.450035 −0.225018 − 0.974355i \(-0.572244\pi\)
−0.225018 + 0.974355i \(0.572244\pi\)
\(80\) 0 0
\(81\) −0.500000 − 0.866025i −0.0555556 − 0.0962250i
\(82\) 1.50000 − 2.59808i 0.165647 − 0.286910i
\(83\) 6.00000 0.658586 0.329293 − 0.944228i \(-0.393190\pi\)
0.329293 + 0.944228i \(0.393190\pi\)
\(84\) −1.00000 + 1.73205i −0.109109 + 0.188982i
\(85\) 0 0
\(86\) 10.0000 1.07833
\(87\) 1.50000 − 2.59808i 0.160817 − 0.278543i
\(88\) −3.00000 − 5.19615i −0.319801 − 0.553912i
\(89\) −9.00000 − 15.5885i −0.953998 − 1.65237i −0.736644 − 0.676280i \(-0.763591\pi\)
−0.217354 − 0.976093i \(-0.569742\pi\)
\(90\) 0 0
\(91\) 2.00000 − 6.92820i 0.209657 − 0.726273i
\(92\) 6.00000 0.625543
\(93\) −2.00000 − 3.46410i −0.207390 − 0.359211i
\(94\) 3.00000 + 5.19615i 0.309426 + 0.535942i
\(95\) 0 0
\(96\) −1.00000 −0.102062
\(97\) 7.00000 − 12.1244i 0.710742 − 1.23104i −0.253837 − 0.967247i \(-0.581693\pi\)
0.964579 − 0.263795i \(-0.0849741\pi\)
\(98\) 1.50000 − 2.59808i 0.151523 − 0.262445i
\(99\) 6.00000 0.603023
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 1950.2.i.m.601.1 2
5.2 odd 4 1950.2.z.g.1849.2 4
5.3 odd 4 1950.2.z.g.1849.1 4
5.4 even 2 78.2.e.a.55.1 ✓ 2
13.9 even 3 inner 1950.2.i.m.451.1 2
15.14 odd 2 234.2.h.a.55.1 2
20.19 odd 2 624.2.q.g.289.1 2
60.59 even 2 1872.2.t.c.289.1 2
65.4 even 6 1014.2.e.a.529.1 2
65.9 even 6 78.2.e.a.61.1 yes 2
65.19 odd 12 1014.2.i.b.823.1 4
65.22 odd 12 1950.2.z.g.1699.1 4
65.24 odd 12 1014.2.b.c.337.1 2
65.29 even 6 1014.2.a.c.1.1 1
65.34 odd 4 1014.2.i.b.361.1 4
65.44 odd 4 1014.2.i.b.361.2 4
65.48 odd 12 1950.2.z.g.1699.2 4
65.49 even 6 1014.2.a.f.1.1 1
65.54 odd 12 1014.2.b.c.337.2 2
65.59 odd 12 1014.2.i.b.823.2 4
65.64 even 2 1014.2.e.a.991.1 2
195.29 odd 6 3042.2.a.i.1.1 1
195.74 odd 6 234.2.h.a.217.1 2
195.89 even 12 3042.2.b.h.1351.2 2
195.119 even 12 3042.2.b.h.1351.1 2
195.179 odd 6 3042.2.a.h.1.1 1
260.139 odd 6 624.2.q.g.529.1 2
260.159 odd 6 8112.2.a.m.1.1 1
260.179 odd 6 8112.2.a.c.1.1 1
780.659 even 6 1872.2.t.c.1153.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
78.2.e.a.55.1 ✓ 2 5.4 even 2
78.2.e.a.61.1 yes 2 65.9 even 6
234.2.h.a.55.1 2 15.14 odd 2
234.2.h.a.217.1 2 195.74 odd 6
624.2.q.g.289.1 2 20.19 odd 2
624.2.q.g.529.1 2 260.139 odd 6
1014.2.a.c.1.1 1 65.29 even 6
1014.2.a.f.1.1 1 65.49 even 6
1014.2.b.c.337.1 2 65.24 odd 12
1014.2.b.c.337.2 2 65.54 odd 12
1014.2.e.a.529.1 2 65.4 even 6
1014.2.e.a.991.1 2 65.64 even 2
1014.2.i.b.361.1 4 65.34 odd 4
1014.2.i.b.361.2 4 65.44 odd 4
1014.2.i.b.823.1 4 65.19 odd 12
1014.2.i.b.823.2 4 65.59 odd 12
1872.2.t.c.289.1 2 60.59 even 2
1872.2.t.c.1153.1 2 780.659 even 6
1950.2.i.m.451.1 2 13.9 even 3 inner
1950.2.i.m.601.1 2 1.1 even 1 trivial
1950.2.z.g.1699.1 4 65.22 odd 12
1950.2.z.g.1699.2 4 65.48 odd 12
1950.2.z.g.1849.1 4 5.3 odd 4
1950.2.z.g.1849.2 4 5.2 odd 4
3042.2.a.h.1.1 1 195.179 odd 6
3042.2.a.i.1.1 1 195.29 odd 6
3042.2.b.h.1351.1 2 195.119 even 12
3042.2.b.h.1351.2 2 195.89 even 12
8112.2.a.c.1.1 1 260.179 odd 6
8112.2.a.m.1.1 1 260.159 odd 6