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Properties

Label 1950.2.y.b.199.1
Level $1950$
Weight $2$
Character 1950.199
Analytic conductor $15.571$
Analytic rank $0$
Dimension $4$
Inner twists $2$

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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(49,1950)] 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("1950.49"); 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, 3, 5])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1950 = 2 \cdot 3 \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1950.y (of order \(6\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,-2,0,-2,0,0,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(15.5708283941\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 78)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 199.1
Root \(-0.866025 - 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 1950.199
Dual form 1950.2.y.b.49.1

$q$-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} +(-0.866025 - 0.500000i) q^{3} +(-0.500000 - 0.866025i) q^{4} +(0.866025 - 0.500000i) q^{6} +(-1.36603 - 2.36603i) q^{7} +1.00000 q^{8} +(0.500000 + 0.866025i) q^{9} +(1.09808 + 0.633975i) q^{11} +1.00000i q^{12} +(-2.50000 - 2.59808i) q^{13} +2.73205 q^{14} +(-0.500000 + 0.866025i) q^{16} +(-4.96410 + 2.86603i) q^{17} -1.00000 q^{18} +(-4.09808 + 2.36603i) q^{19} +2.73205i q^{21} +(-1.09808 + 0.633975i) q^{22} +(3.63397 + 2.09808i) q^{23} +(-0.866025 - 0.500000i) q^{24} +(3.50000 - 0.866025i) q^{26} -1.00000i q^{27} +(-1.36603 + 2.36603i) q^{28} +(-2.23205 + 3.86603i) q^{29} +1.46410i q^{31} +(-0.500000 - 0.866025i) q^{32} +(-0.633975 - 1.09808i) q^{33} -5.73205i q^{34} +(0.500000 - 0.866025i) q^{36} +(1.76795 - 3.06218i) q^{37} -4.73205i q^{38} +(0.866025 + 3.50000i) q^{39} +(8.13397 + 4.69615i) q^{41} +(-2.36603 - 1.36603i) q^{42} +(8.36603 - 4.83013i) q^{43} -1.26795i q^{44} +(-3.63397 + 2.09808i) q^{46} -2.19615 q^{47} +(0.866025 - 0.500000i) q^{48} +(-0.232051 + 0.401924i) q^{49} +5.73205 q^{51} +(-1.00000 + 3.46410i) q^{52} -6.46410i q^{53} +(0.866025 + 0.500000i) q^{54} +(-1.36603 - 2.36603i) q^{56} +4.73205 q^{57} +(-2.23205 - 3.86603i) q^{58} +(6.92820 - 4.00000i) q^{59} +(4.59808 + 7.96410i) q^{61} +(-1.26795 - 0.732051i) q^{62} +(1.36603 - 2.36603i) q^{63} +1.00000 q^{64} +1.26795 q^{66} +(-6.56218 + 11.3660i) q^{67} +(4.96410 + 2.86603i) q^{68} +(-2.09808 - 3.63397i) q^{69} +(4.09808 - 2.36603i) q^{71} +(0.500000 + 0.866025i) q^{72} -6.26795 q^{73} +(1.76795 + 3.06218i) q^{74} +(4.09808 + 2.36603i) q^{76} -3.46410i q^{77} +(-3.46410 - 1.00000i) q^{78} +2.53590 q^{79} +(-0.500000 + 0.866025i) q^{81} +(-8.13397 + 4.69615i) q^{82} +0.196152 q^{83} +(2.36603 - 1.36603i) q^{84} +9.66025i q^{86} +(3.86603 - 2.23205i) q^{87} +(1.09808 + 0.633975i) q^{88} +(8.19615 + 4.73205i) q^{89} +(-2.73205 + 9.46410i) q^{91} -4.19615i q^{92} +(0.732051 - 1.26795i) q^{93} +(1.09808 - 1.90192i) q^{94} +1.00000i q^{96} +(3.00000 + 5.19615i) q^{97} +(-0.232051 - 0.401924i) q^{98} +1.26795i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{2} - 2 q^{4} - 2 q^{7} + 4 q^{8} + 2 q^{9} - 6 q^{11} - 10 q^{13} + 4 q^{14} - 2 q^{16} - 6 q^{17} - 4 q^{18} - 6 q^{19} + 6 q^{22} + 18 q^{23} + 14 q^{26} - 2 q^{28} - 2 q^{29} - 2 q^{32} - 6 q^{33}+ \cdots + 6 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(301\) \(1301\) \(1327\)
\(\chi(n)\) \(e\left(\frac{1}{6}\right)\) \(1\) \(-1\)

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
\(3\) −0.866025 − 0.500000i −0.500000 − 0.288675i
\(4\) −0.500000 − 0.866025i −0.250000 − 0.433013i
\(5\) 0 0
\(6\) 0.866025 − 0.500000i 0.353553 − 0.204124i
\(7\) −1.36603 − 2.36603i −0.516309 − 0.894274i −0.999821 − 0.0189356i \(-0.993972\pi\)
0.483512 − 0.875338i \(-0.339361\pi\)
\(8\) 1.00000 0.353553
\(9\) 0.500000 + 0.866025i 0.166667 + 0.288675i
\(10\) 0 0
\(11\) 1.09808 + 0.633975i 0.331082 + 0.191151i 0.656322 − 0.754481i \(-0.272111\pi\)
−0.325239 + 0.945632i \(0.605445\pi\)
\(12\) 1.00000i 0.288675i
\(13\) −2.50000 − 2.59808i −0.693375 − 0.720577i
\(14\) 2.73205 0.730171
\(15\) 0 0
\(16\) −0.500000 + 0.866025i −0.125000 + 0.216506i
\(17\) −4.96410 + 2.86603i −1.20397 + 0.695113i −0.961436 − 0.275029i \(-0.911312\pi\)
−0.242536 + 0.970143i \(0.577979\pi\)
\(18\) −1.00000 −0.235702
\(19\) −4.09808 + 2.36603i −0.940163 + 0.542803i −0.890011 − 0.455938i \(-0.849304\pi\)
−0.0501517 + 0.998742i \(0.515970\pi\)
\(20\) 0 0
\(21\) 2.73205i 0.596182i
\(22\) −1.09808 + 0.633975i −0.234111 + 0.135164i
\(23\) 3.63397 + 2.09808i 0.757736 + 0.437479i 0.828482 − 0.560015i \(-0.189205\pi\)
−0.0707462 + 0.997494i \(0.522538\pi\)
\(24\) −0.866025 − 0.500000i −0.176777 − 0.102062i
\(25\) 0 0
\(26\) 3.50000 − 0.866025i 0.686406 − 0.169842i
\(27\) − 1.00000i − 0.192450i
\(28\) −1.36603 + 2.36603i −0.258155 + 0.447137i
\(29\) −2.23205 + 3.86603i −0.414481 + 0.717903i −0.995374 − 0.0960774i \(-0.969370\pi\)
0.580892 + 0.813980i \(0.302704\pi\)
\(30\) 0 0
\(31\) 1.46410i 0.262960i 0.991319 + 0.131480i \(0.0419730\pi\)
−0.991319 + 0.131480i \(0.958027\pi\)
\(32\) −0.500000 − 0.866025i −0.0883883 − 0.153093i
\(33\) −0.633975 − 1.09808i −0.110361 − 0.191151i
\(34\) − 5.73205i − 0.983039i
\(35\) 0 0
\(36\) 0.500000 − 0.866025i 0.0833333 − 0.144338i
\(37\) 1.76795 − 3.06218i 0.290649 − 0.503419i −0.683314 − 0.730124i \(-0.739462\pi\)
0.973963 + 0.226705i \(0.0727955\pi\)
\(38\) − 4.73205i − 0.767640i
\(39\) 0.866025 + 3.50000i 0.138675 + 0.560449i
\(40\) 0 0
\(41\) 8.13397 + 4.69615i 1.27031 + 0.733416i 0.975047 − 0.221999i \(-0.0712582\pi\)
0.295267 + 0.955415i \(0.404592\pi\)
\(42\) −2.36603 − 1.36603i −0.365086 − 0.210782i
\(43\) 8.36603 − 4.83013i 1.27581 − 0.736587i 0.299732 − 0.954023i \(-0.403103\pi\)
0.976075 + 0.217436i \(0.0697693\pi\)
\(44\) − 1.26795i − 0.191151i
\(45\) 0 0
\(46\) −3.63397 + 2.09808i −0.535800 + 0.309344i
\(47\) −2.19615 −0.320342 −0.160171 − 0.987089i \(-0.551205\pi\)
−0.160171 + 0.987089i \(0.551205\pi\)
\(48\) 0.866025 − 0.500000i 0.125000 − 0.0721688i
\(49\) −0.232051 + 0.401924i −0.0331501 + 0.0574177i
\(50\) 0 0
\(51\) 5.73205 0.802648
\(52\) −1.00000 + 3.46410i −0.138675 + 0.480384i
\(53\) − 6.46410i − 0.887913i −0.896048 − 0.443956i \(-0.853575\pi\)
0.896048 − 0.443956i \(-0.146425\pi\)
\(54\) 0.866025 + 0.500000i 0.117851 + 0.0680414i
\(55\) 0 0
\(56\) −1.36603 − 2.36603i −0.182543 − 0.316173i
\(57\) 4.73205 0.626775
\(58\) −2.23205 − 3.86603i −0.293083 − 0.507634i
\(59\) 6.92820 − 4.00000i 0.901975 − 0.520756i 0.0241347 − 0.999709i \(-0.492317\pi\)
0.877841 + 0.478953i \(0.158984\pi\)
\(60\) 0 0
\(61\) 4.59808 + 7.96410i 0.588723 + 1.01970i 0.994400 + 0.105682i \(0.0337026\pi\)
−0.405677 + 0.914017i \(0.632964\pi\)
\(62\) −1.26795 − 0.732051i −0.161030 − 0.0929705i
\(63\) 1.36603 − 2.36603i 0.172103 − 0.298091i
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 1.26795 0.156074
\(67\) −6.56218 + 11.3660i −0.801698 + 1.38858i 0.116800 + 0.993155i \(0.462736\pi\)
−0.918498 + 0.395426i \(0.870597\pi\)
\(68\) 4.96410 + 2.86603i 0.601986 + 0.347557i
\(69\) −2.09808 − 3.63397i −0.252579 − 0.437479i
\(70\) 0 0
\(71\) 4.09808 − 2.36603i 0.486352 − 0.280796i −0.236708 − 0.971581i \(-0.576068\pi\)
0.723060 + 0.690785i \(0.242735\pi\)
\(72\) 0.500000 + 0.866025i 0.0589256 + 0.102062i
\(73\) −6.26795 −0.733608 −0.366804 − 0.930298i \(-0.619548\pi\)
−0.366804 + 0.930298i \(0.619548\pi\)
\(74\) 1.76795 + 3.06218i 0.205520 + 0.355971i
\(75\) 0 0
\(76\) 4.09808 + 2.36603i 0.470082 + 0.271402i
\(77\) − 3.46410i − 0.394771i
\(78\) −3.46410 − 1.00000i −0.392232 − 0.113228i
\(79\) 2.53590 0.285311 0.142655 − 0.989772i \(-0.454436\pi\)
0.142655 + 0.989772i \(0.454436\pi\)
\(80\) 0 0
\(81\) −0.500000 + 0.866025i −0.0555556 + 0.0962250i
\(82\) −8.13397 + 4.69615i −0.898247 + 0.518603i
\(83\) 0.196152 0.0215305 0.0107653 − 0.999942i \(-0.496573\pi\)
0.0107653 + 0.999942i \(0.496573\pi\)
\(84\) 2.36603 − 1.36603i 0.258155 − 0.149046i
\(85\) 0 0
\(86\) 9.66025i 1.04169i
\(87\) 3.86603 − 2.23205i 0.414481 − 0.239301i
\(88\) 1.09808 + 0.633975i 0.117055 + 0.0675819i
\(89\) 8.19615 + 4.73205i 0.868790 + 0.501596i 0.866946 − 0.498402i \(-0.166080\pi\)
0.00184433 + 0.999998i \(0.499413\pi\)
\(90\) 0 0
\(91\) −2.73205 + 9.46410i −0.286397 + 0.992107i
\(92\) − 4.19615i − 0.437479i
\(93\) 0.732051 − 1.26795i 0.0759101 − 0.131480i
\(94\) 1.09808 − 1.90192i 0.113258 − 0.196168i
\(95\) 0 0
\(96\) 1.00000i 0.102062i
\(97\) 3.00000 + 5.19615i 0.304604 + 0.527589i 0.977173 − 0.212445i \(-0.0681426\pi\)
−0.672569 + 0.740034i \(0.734809\pi\)
\(98\) −0.232051 − 0.401924i −0.0234407 − 0.0406004i
\(99\) 1.26795i 0.127434i
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.y.b.199.1 4
5.2 odd 4 78.2.i.a.43.1 ✓ 4
5.3 odd 4 1950.2.bc.d.901.2 4
5.4 even 2 1950.2.y.g.199.2 4
13.10 even 6 1950.2.y.g.49.2 4
15.2 even 4 234.2.l.c.199.2 4
20.7 even 4 624.2.bv.e.433.2 4
60.47 odd 4 1872.2.by.h.433.1 4
65.2 even 12 1014.2.e.g.991.2 4
65.7 even 12 1014.2.a.i.1.1 2
65.12 odd 4 1014.2.i.a.823.2 4
65.17 odd 12 1014.2.b.e.337.1 4
65.22 odd 12 1014.2.b.e.337.4 4
65.23 odd 12 1950.2.bc.d.751.2 4
65.32 even 12 1014.2.a.k.1.2 2
65.37 even 12 1014.2.e.i.991.1 4
65.42 odd 12 1014.2.i.a.361.2 4
65.47 even 4 1014.2.e.i.529.1 4
65.49 even 6 inner 1950.2.y.b.49.1 4
65.57 even 4 1014.2.e.g.529.2 4
65.62 odd 12 78.2.i.a.49.1 yes 4
195.17 even 12 3042.2.b.i.1351.4 4
195.32 odd 12 3042.2.a.p.1.1 2
195.62 even 12 234.2.l.c.127.2 4
195.137 odd 12 3042.2.a.y.1.2 2
195.152 even 12 3042.2.b.i.1351.1 4
260.7 odd 12 8112.2.a.bj.1.1 2
260.127 even 12 624.2.bv.e.49.1 4
260.227 odd 12 8112.2.a.bp.1.2 2
780.647 odd 12 1872.2.by.h.1297.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
78.2.i.a.43.1 ✓ 4 5.2 odd 4
78.2.i.a.49.1 yes 4 65.62 odd 12
234.2.l.c.127.2 4 195.62 even 12
234.2.l.c.199.2 4 15.2 even 4
624.2.bv.e.49.1 4 260.127 even 12
624.2.bv.e.433.2 4 20.7 even 4
1014.2.a.i.1.1 2 65.7 even 12
1014.2.a.k.1.2 2 65.32 even 12
1014.2.b.e.337.1 4 65.17 odd 12
1014.2.b.e.337.4 4 65.22 odd 12
1014.2.e.g.529.2 4 65.57 even 4
1014.2.e.g.991.2 4 65.2 even 12
1014.2.e.i.529.1 4 65.47 even 4
1014.2.e.i.991.1 4 65.37 even 12
1014.2.i.a.361.2 4 65.42 odd 12
1014.2.i.a.823.2 4 65.12 odd 4
1872.2.by.h.433.1 4 60.47 odd 4
1872.2.by.h.1297.2 4 780.647 odd 12
1950.2.y.b.49.1 4 65.49 even 6 inner
1950.2.y.b.199.1 4 1.1 even 1 trivial
1950.2.y.g.49.2 4 13.10 even 6
1950.2.y.g.199.2 4 5.4 even 2
1950.2.bc.d.751.2 4 65.23 odd 12
1950.2.bc.d.901.2 4 5.3 odd 4
3042.2.a.p.1.1 2 195.32 odd 12
3042.2.a.y.1.2 2 195.137 odd 12
3042.2.b.i.1351.1 4 195.152 even 12
3042.2.b.i.1351.4 4 195.17 even 12
8112.2.a.bj.1.1 2 260.7 odd 12
8112.2.a.bp.1.2 2 260.227 odd 12