Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [185,2,Mod(68,185)] 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("185.68"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(185, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([3, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 185 = 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 185.k (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,-2,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(1.47723243739\)
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: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 68.1
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 185.68
Dual form 185.2.k.b.117.1

$q$-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 q^{2} +(1.00000 - 1.00000i) q^{3} -1.00000 q^{4} +(-1.00000 + 2.00000i) q^{5} +(-1.00000 + 1.00000i) q^{6} +(-3.00000 + 3.00000i) q^{7} +3.00000 q^{8} +1.00000i q^{9} +(1.00000 - 2.00000i) q^{10} +2.00000i q^{11} +(-1.00000 + 1.00000i) q^{12} +2.00000 q^{13} +(3.00000 - 3.00000i) q^{14} +(1.00000 + 3.00000i) q^{15} -1.00000 q^{16} -4.00000i q^{17} -1.00000i q^{18} +(3.00000 + 3.00000i) q^{19} +(1.00000 - 2.00000i) q^{20} +6.00000i q^{21} -2.00000i q^{22} -8.00000 q^{23} +(3.00000 - 3.00000i) q^{24} +(-3.00000 - 4.00000i) q^{25} -2.00000 q^{26} +(4.00000 + 4.00000i) q^{27} +(3.00000 - 3.00000i) q^{28} +(-7.00000 + 7.00000i) q^{29} +(-1.00000 - 3.00000i) q^{30} +(3.00000 + 3.00000i) q^{31} -5.00000 q^{32} +(2.00000 + 2.00000i) q^{33} +4.00000i q^{34} +(-3.00000 - 9.00000i) q^{35} -1.00000i q^{36} +(-1.00000 - 6.00000i) q^{37} +(-3.00000 - 3.00000i) q^{38} +(2.00000 - 2.00000i) q^{39} +(-3.00000 + 6.00000i) q^{40} -6.00000i q^{42} +12.0000 q^{43} -2.00000i q^{44} +(-2.00000 - 1.00000i) q^{45} +8.00000 q^{46} +(5.00000 - 5.00000i) q^{47} +(-1.00000 + 1.00000i) q^{48} -11.0000i q^{49} +(3.00000 + 4.00000i) q^{50} +(-4.00000 - 4.00000i) q^{51} -2.00000 q^{52} +(-3.00000 - 3.00000i) q^{53} +(-4.00000 - 4.00000i) q^{54} +(-4.00000 - 2.00000i) q^{55} +(-9.00000 + 9.00000i) q^{56} +6.00000 q^{57} +(7.00000 - 7.00000i) q^{58} +(7.00000 + 7.00000i) q^{59} +(-1.00000 - 3.00000i) q^{60} +(1.00000 + 1.00000i) q^{61} +(-3.00000 - 3.00000i) q^{62} +(-3.00000 - 3.00000i) q^{63} +7.00000 q^{64} +(-2.00000 + 4.00000i) q^{65} +(-2.00000 - 2.00000i) q^{66} +(3.00000 + 3.00000i) q^{67} +4.00000i q^{68} +(-8.00000 + 8.00000i) q^{69} +(3.00000 + 9.00000i) q^{70} +8.00000 q^{71} +3.00000i q^{72} +(1.00000 - 1.00000i) q^{73} +(1.00000 + 6.00000i) q^{74} +(-7.00000 - 1.00000i) q^{75} +(-3.00000 - 3.00000i) q^{76} +(-6.00000 - 6.00000i) q^{77} +(-2.00000 + 2.00000i) q^{78} +(3.00000 + 3.00000i) q^{79} +(1.00000 - 2.00000i) q^{80} +5.00000 q^{81} +(-5.00000 - 5.00000i) q^{83} -6.00000i q^{84} +(8.00000 + 4.00000i) q^{85} -12.0000 q^{86} +14.0000i q^{87} +6.00000i q^{88} +(5.00000 - 5.00000i) q^{89} +(2.00000 + 1.00000i) q^{90} +(-6.00000 + 6.00000i) q^{91} +8.00000 q^{92} +6.00000 q^{93} +(-5.00000 + 5.00000i) q^{94} +(-9.00000 + 3.00000i) q^{95} +(-5.00000 + 5.00000i) q^{96} +8.00000i q^{97} +11.0000i q^{98} -2.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + 2 q^{3} - 2 q^{4} - 2 q^{5} - 2 q^{6} - 6 q^{7} + 6 q^{8} + 2 q^{10} - 2 q^{12} + 4 q^{13} + 6 q^{14} + 2 q^{15} - 2 q^{16} + 6 q^{19} + 2 q^{20} - 16 q^{23} + 6 q^{24} - 6 q^{25} - 4 q^{26}+ \cdots - 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/185\mathbb{Z}\right)^\times\).

\(n\) \(76\) \(112\)
\(\chi(n)\) \(e\left(\frac{1}{4}\right)\) \(e\left(\frac{3}{4}\right)\)

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 −0.707107 −0.353553 − 0.935414i \(-0.615027\pi\)
−0.353553 + 0.935414i \(0.615027\pi\)
\(3\) 1.00000 − 1.00000i 0.577350 − 0.577350i −0.356822 − 0.934172i \(-0.616140\pi\)
0.934172 + 0.356822i \(0.116140\pi\)
\(4\) −1.00000 −0.500000
\(5\) −1.00000 + 2.00000i −0.447214 + 0.894427i
\(6\) −1.00000 + 1.00000i −0.408248 + 0.408248i
\(7\) −3.00000 + 3.00000i −1.13389 + 1.13389i −0.144370 + 0.989524i \(0.546115\pi\)
−0.989524 + 0.144370i \(0.953885\pi\)
\(8\) 3.00000 1.06066
\(9\) 1.00000i 0.333333i
\(10\) 1.00000 − 2.00000i 0.316228 − 0.632456i
\(11\) 2.00000i 0.603023i 0.953463 + 0.301511i \(0.0974911\pi\)
−0.953463 + 0.301511i \(0.902509\pi\)
\(12\) −1.00000 + 1.00000i −0.288675 + 0.288675i
\(13\) 2.00000 0.554700 0.277350 − 0.960769i \(-0.410544\pi\)
0.277350 + 0.960769i \(0.410544\pi\)
\(14\) 3.00000 − 3.00000i 0.801784 − 0.801784i
\(15\) 1.00000 + 3.00000i 0.258199 + 0.774597i
\(16\) −1.00000 −0.250000
\(17\) − 4.00000i − 0.970143i −0.874475 − 0.485071i \(-0.838794\pi\)
0.874475 − 0.485071i \(-0.161206\pi\)
\(18\) − 1.00000i − 0.235702i
\(19\) 3.00000 + 3.00000i 0.688247 + 0.688247i 0.961844 − 0.273597i \(-0.0882135\pi\)
−0.273597 + 0.961844i \(0.588214\pi\)
\(20\) 1.00000 − 2.00000i 0.223607 − 0.447214i
\(21\) 6.00000i 1.30931i
\(22\) − 2.00000i − 0.426401i
\(23\) −8.00000 −1.66812 −0.834058 − 0.551677i \(-0.813988\pi\)
−0.834058 + 0.551677i \(0.813988\pi\)
\(24\) 3.00000 − 3.00000i 0.612372 − 0.612372i
\(25\) −3.00000 − 4.00000i −0.600000 − 0.800000i
\(26\) −2.00000 −0.392232
\(27\) 4.00000 + 4.00000i 0.769800 + 0.769800i
\(28\) 3.00000 − 3.00000i 0.566947 − 0.566947i
\(29\) −7.00000 + 7.00000i −1.29987 + 1.29987i −0.371391 + 0.928477i \(0.621119\pi\)
−0.928477 + 0.371391i \(0.878881\pi\)
\(30\) −1.00000 − 3.00000i −0.182574 − 0.547723i
\(31\) 3.00000 + 3.00000i 0.538816 + 0.538816i 0.923181 − 0.384365i \(-0.125580\pi\)
−0.384365 + 0.923181i \(0.625580\pi\)
\(32\) −5.00000 −0.883883
\(33\) 2.00000 + 2.00000i 0.348155 + 0.348155i
\(34\) 4.00000i 0.685994i
\(35\) −3.00000 − 9.00000i −0.507093 − 1.52128i
\(36\) − 1.00000i − 0.166667i
\(37\) −1.00000 − 6.00000i −0.164399 − 0.986394i
\(38\) −3.00000 − 3.00000i −0.486664 − 0.486664i
\(39\) 2.00000 − 2.00000i 0.320256 − 0.320256i
\(40\) −3.00000 + 6.00000i −0.474342 + 0.948683i
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) − 6.00000i − 0.925820i
\(43\) 12.0000 1.82998 0.914991 − 0.403473i \(-0.132197\pi\)
0.914991 + 0.403473i \(0.132197\pi\)
\(44\) − 2.00000i − 0.301511i
\(45\) −2.00000 − 1.00000i −0.298142 − 0.149071i
\(46\) 8.00000 1.17954
\(47\) 5.00000 − 5.00000i 0.729325 − 0.729325i −0.241160 − 0.970485i \(-0.577528\pi\)
0.970485 + 0.241160i \(0.0775280\pi\)
\(48\) −1.00000 + 1.00000i −0.144338 + 0.144338i
\(49\) − 11.0000i − 1.57143i
\(50\) 3.00000 + 4.00000i 0.424264 + 0.565685i
\(51\) −4.00000 − 4.00000i −0.560112 − 0.560112i
\(52\) −2.00000 −0.277350
\(53\) −3.00000 − 3.00000i −0.412082 − 0.412082i 0.470381 − 0.882463i \(-0.344116\pi\)
−0.882463 + 0.470381i \(0.844116\pi\)
\(54\) −4.00000 − 4.00000i −0.544331 − 0.544331i
\(55\) −4.00000 − 2.00000i −0.539360 − 0.269680i
\(56\) −9.00000 + 9.00000i −1.20268 + 1.20268i
\(57\) 6.00000 0.794719
\(58\) 7.00000 − 7.00000i 0.919145 − 0.919145i
\(59\) 7.00000 + 7.00000i 0.911322 + 0.911322i 0.996376 − 0.0850540i \(-0.0271063\pi\)
−0.0850540 + 0.996376i \(0.527106\pi\)
\(60\) −1.00000 − 3.00000i −0.129099 − 0.387298i
\(61\) 1.00000 + 1.00000i 0.128037 + 0.128037i 0.768221 − 0.640184i \(-0.221142\pi\)
−0.640184 + 0.768221i \(0.721142\pi\)
\(62\) −3.00000 − 3.00000i −0.381000 − 0.381000i
\(63\) −3.00000 − 3.00000i −0.377964 − 0.377964i
\(64\) 7.00000 0.875000
\(65\) −2.00000 + 4.00000i −0.248069 + 0.496139i
\(66\) −2.00000 − 2.00000i −0.246183 − 0.246183i
\(67\) 3.00000 + 3.00000i 0.366508 + 0.366508i 0.866202 − 0.499694i \(-0.166554\pi\)
−0.499694 + 0.866202i \(0.666554\pi\)
\(68\) 4.00000i 0.485071i
\(69\) −8.00000 + 8.00000i −0.963087 + 0.963087i
\(70\) 3.00000 + 9.00000i 0.358569 + 1.07571i
\(71\) 8.00000 0.949425 0.474713 − 0.880141i \(-0.342552\pi\)
0.474713 + 0.880141i \(0.342552\pi\)
\(72\) 3.00000i 0.353553i
\(73\) 1.00000 − 1.00000i 0.117041 − 0.117041i −0.646160 − 0.763202i \(-0.723626\pi\)
0.763202 + 0.646160i \(0.223626\pi\)
\(74\) 1.00000 + 6.00000i 0.116248 + 0.697486i
\(75\) −7.00000 − 1.00000i −0.808290 − 0.115470i
\(76\) −3.00000 − 3.00000i −0.344124 − 0.344124i
\(77\) −6.00000 − 6.00000i −0.683763 − 0.683763i
\(78\) −2.00000 + 2.00000i −0.226455 + 0.226455i
\(79\) 3.00000 + 3.00000i 0.337526 + 0.337526i 0.855436 − 0.517909i \(-0.173290\pi\)
−0.517909 + 0.855436i \(0.673290\pi\)
\(80\) 1.00000 − 2.00000i 0.111803 − 0.223607i
\(81\) 5.00000 0.555556
\(82\) 0 0
\(83\) −5.00000 − 5.00000i −0.548821 − 0.548821i 0.377279 − 0.926100i \(-0.376860\pi\)
−0.926100 + 0.377279i \(0.876860\pi\)
\(84\) − 6.00000i − 0.654654i
\(85\) 8.00000 + 4.00000i 0.867722 + 0.433861i
\(86\) −12.0000 −1.29399
\(87\) 14.0000i 1.50096i
\(88\) 6.00000i 0.639602i
\(89\) 5.00000 − 5.00000i 0.529999 − 0.529999i −0.390573 − 0.920572i \(-0.627723\pi\)
0.920572 + 0.390573i \(0.127723\pi\)
\(90\) 2.00000 + 1.00000i 0.210819 + 0.105409i
\(91\) −6.00000 + 6.00000i −0.628971 + 0.628971i
\(92\) 8.00000 0.834058
\(93\) 6.00000 0.622171
\(94\) −5.00000 + 5.00000i −0.515711 + 0.515711i
\(95\) −9.00000 + 3.00000i −0.923381 + 0.307794i
\(96\) −5.00000 + 5.00000i −0.510310 + 0.510310i
\(97\) 8.00000i 0.812277i 0.913812 + 0.406138i \(0.133125\pi\)
−0.913812 + 0.406138i \(0.866875\pi\)
\(98\) 11.0000i 1.11117i
\(99\) −2.00000 −0.201008
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 185.2.k.b.68.1 yes 2
5.2 odd 4 185.2.f.a.142.1 yes 2
5.3 odd 4 925.2.f.b.882.1 2
5.4 even 2 925.2.k.a.68.1 2
37.6 odd 4 185.2.f.a.43.1 ✓ 2
185.43 even 4 925.2.k.a.857.1 2
185.117 even 4 inner 185.2.k.b.117.1 yes 2
185.154 odd 4 925.2.f.b.43.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
185.2.f.a.43.1 ✓ 2 37.6 odd 4
185.2.f.a.142.1 yes 2 5.2 odd 4
185.2.k.b.68.1 yes 2 1.1 even 1 trivial
185.2.k.b.117.1 yes 2 185.117 even 4 inner
925.2.f.b.43.1 2 185.154 odd 4
925.2.f.b.882.1 2 5.3 odd 4
925.2.k.a.68.1 2 5.4 even 2
925.2.k.a.857.1 2 185.43 even 4