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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [185,2,Mod(43,185)] 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("185.43"); 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, 3])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 185 = 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 185.f (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,-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: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 142.1
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 185.142
Dual form 185.2.f.a.43.1

$q$-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-1.00000i q^{2} +(-1.00000 - 1.00000i) q^{3} +1.00000 q^{4} +(-2.00000 + 1.00000i) q^{5} +(-1.00000 + 1.00000i) q^{6} +(-3.00000 - 3.00000i) q^{7} -3.00000i q^{8} -1.00000i q^{9} +(1.00000 + 2.00000i) q^{10} +2.00000i q^{11} +(-1.00000 - 1.00000i) q^{12} -2.00000i q^{13} +(-3.00000 + 3.00000i) q^{14} +(3.00000 + 1.00000i) q^{15} -1.00000 q^{16} +4.00000 q^{17} -1.00000 q^{18} +(-3.00000 - 3.00000i) q^{19} +(-2.00000 + 1.00000i) q^{20} +6.00000i q^{21} +2.00000 q^{22} +8.00000i 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} -5.00000i q^{32} +(2.00000 - 2.00000i) q^{33} -4.00000i q^{34} +(9.00000 + 3.00000i) q^{35} -1.00000i q^{36} +(6.00000 - 1.00000i) q^{37} +(-3.00000 + 3.00000i) q^{38} +(-2.00000 + 2.00000i) q^{39} +(3.00000 + 6.00000i) q^{40} +6.00000 q^{42} -12.0000i q^{43} +2.00000i q^{44} +(1.00000 + 2.00000i) q^{45} +8.00000 q^{46} +(5.00000 + 5.00000i) q^{47} +(1.00000 + 1.00000i) q^{48} +11.0000i q^{49} +(-4.00000 - 3.00000i) q^{50} +(-4.00000 - 4.00000i) q^{51} -2.00000i q^{52} +(-3.00000 + 3.00000i) q^{53} +(4.00000 + 4.00000i) q^{54} +(-2.00000 - 4.00000i) q^{55} +(-9.00000 + 9.00000i) q^{56} +6.00000i q^{57} +(-7.00000 - 7.00000i) q^{58} +(-7.00000 - 7.00000i) q^{59} +(3.00000 + 1.00000i) 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} +4.00000 q^{68} +(8.00000 - 8.00000i) q^{69} +(3.00000 - 9.00000i) q^{70} +8.00000 q^{71} -3.00000 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} +(2.00000 - 1.00000i) 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} -14.0000 q^{87} +6.00000 q^{88} +(-5.00000 + 5.00000i) q^{89} +(2.00000 - 1.00000i) q^{90} +(-6.00000 + 6.00000i) q^{91} +8.00000i q^{92} -6.00000i q^{93} +(5.00000 - 5.00000i) q^{94} +(9.00000 + 3.00000i) q^{95} +(-5.00000 + 5.00000i) q^{96} -8.00000 q^{97} +11.0000 q^{98} +2.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{3} + 2 q^{4} - 4 q^{5} - 2 q^{6} - 6 q^{7} + 2 q^{10} - 2 q^{12} - 6 q^{14} + 6 q^{15} - 2 q^{16} + 8 q^{17} - 2 q^{18} - 6 q^{19} - 4 q^{20} + 4 q^{22} - 6 q^{24} + 6 q^{25} - 4 q^{26} - 8 q^{27}+ \cdots + 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 1.00000i − 0.707107i −0.935414 − 0.353553i \(-0.884973\pi\)
0.935414 − 0.353553i \(-0.115027\pi\)
\(3\) −1.00000 − 1.00000i −0.577350 − 0.577350i 0.356822 − 0.934172i \(-0.383860\pi\)
−0.934172 + 0.356822i \(0.883860\pi\)
\(4\) 1.00000 0.500000
\(5\) −2.00000 + 1.00000i −0.894427 + 0.447214i
\(6\) −1.00000 + 1.00000i −0.408248 + 0.408248i
\(7\) −3.00000 − 3.00000i −1.13389 − 1.13389i −0.989524 − 0.144370i \(-0.953885\pi\)
−0.144370 − 0.989524i \(-0.546115\pi\)
\(8\) − 3.00000i − 1.06066i
\(9\) − 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
\(13\) − 2.00000i − 0.554700i −0.960769 − 0.277350i \(-0.910544\pi\)
0.960769 − 0.277350i \(-0.0894562\pi\)
\(14\) −3.00000 + 3.00000i −0.801784 + 0.801784i
\(15\) 3.00000 + 1.00000i 0.774597 + 0.258199i
\(16\) −1.00000 −0.250000
\(17\) 4.00000 0.970143 0.485071 − 0.874475i \(-0.338794\pi\)
0.485071 + 0.874475i \(0.338794\pi\)
\(18\) −1.00000 −0.235702
\(19\) −3.00000 − 3.00000i −0.688247 − 0.688247i 0.273597 − 0.961844i \(-0.411786\pi\)
−0.961844 + 0.273597i \(0.911786\pi\)
\(20\) −2.00000 + 1.00000i −0.447214 + 0.223607i
\(21\) 6.00000i 1.30931i
\(22\) 2.00000 0.426401
\(23\) 8.00000i 1.66812i 0.551677 + 0.834058i \(0.313988\pi\)
−0.551677 + 0.834058i \(0.686012\pi\)
\(24\) −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 \(-0.378881\pi\)
0.928477 − 0.371391i \(-0.121119\pi\)
\(30\) 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 \(0.625580\pi\)
\(32\) − 5.00000i − 0.883883i
\(33\) 2.00000 − 2.00000i 0.348155 − 0.348155i
\(34\) − 4.00000i − 0.685994i
\(35\) 9.00000 + 3.00000i 1.52128 + 0.507093i
\(36\) − 1.00000i − 0.166667i
\(37\) 6.00000 − 1.00000i 0.986394 − 0.164399i
\(38\) −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 \(\pi\)
\(42\) 6.00000 0.925820
\(43\) − 12.0000i − 1.82998i −0.403473 − 0.914991i \(-0.632197\pi\)
0.403473 − 0.914991i \(-0.367803\pi\)
\(44\) 2.00000i 0.301511i
\(45\) 1.00000 + 2.00000i 0.149071 + 0.298142i
\(46\) 8.00000 1.17954
\(47\) 5.00000 + 5.00000i 0.729325 + 0.729325i 0.970485 − 0.241160i \(-0.0775280\pi\)
−0.241160 + 0.970485i \(0.577528\pi\)
\(48\) 1.00000 + 1.00000i 0.144338 + 0.144338i
\(49\) 11.0000i 1.57143i
\(50\) −4.00000 − 3.00000i −0.565685 − 0.424264i
\(51\) −4.00000 − 4.00000i −0.560112 − 0.560112i
\(52\) − 2.00000i − 0.277350i
\(53\) −3.00000 + 3.00000i −0.412082 + 0.412082i −0.882463 − 0.470381i \(-0.844116\pi\)
0.470381 + 0.882463i \(0.344116\pi\)
\(54\) 4.00000 + 4.00000i 0.544331 + 0.544331i
\(55\) −2.00000 − 4.00000i −0.269680 − 0.539360i
\(56\) −9.00000 + 9.00000i −1.20268 + 1.20268i
\(57\) 6.00000i 0.794719i
\(58\) −7.00000 − 7.00000i −0.919145 − 0.919145i
\(59\) −7.00000 − 7.00000i −0.911322 − 0.911322i 0.0850540 − 0.996376i \(-0.472894\pi\)
−0.996376 + 0.0850540i \(0.972894\pi\)
\(60\) 3.00000 + 1.00000i 0.387298 + 0.129099i
\(61\) 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 \(-0.833446\pi\)
0.499694 + 0.866202i \(0.333446\pi\)
\(68\) 4.00000 0.485071
\(69\) 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 \(0.342552\pi\)
\(72\) −3.00000 −0.353553
\(73\) −1.00000 − 1.00000i −0.117041 − 0.117041i 0.646160 − 0.763202i \(-0.276374\pi\)
−0.763202 + 0.646160i \(0.776374\pi\)
\(74\) −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 0.517909 − 0.855436i \(-0.326710\pi\)
−0.855436 + 0.517909i \(0.826710\pi\)
\(80\) 2.00000 − 1.00000i 0.223607 − 0.111803i
\(81\) 5.00000 0.555556
\(82\) 0 0
\(83\) −5.00000 + 5.00000i −0.548821 + 0.548821i −0.926100 − 0.377279i \(-0.876860\pi\)
0.377279 + 0.926100i \(0.376860\pi\)
\(84\) 6.00000i 0.654654i
\(85\) −8.00000 + 4.00000i −0.867722 + 0.433861i
\(86\) −12.0000 −1.29399
\(87\) −14.0000 −1.50096
\(88\) 6.00000 0.639602
\(89\) −5.00000 + 5.00000i −0.529999 + 0.529999i −0.920572 − 0.390573i \(-0.872277\pi\)
0.390573 + 0.920572i \(0.372277\pi\)
\(90\) 2.00000 − 1.00000i 0.210819 − 0.105409i
\(91\) −6.00000 + 6.00000i −0.628971 + 0.628971i
\(92\) 8.00000i 0.834058i
\(93\) − 6.00000i − 0.622171i
\(94\) 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
\(97\) −8.00000 −0.812277 −0.406138 − 0.913812i \(-0.633125\pi\)
−0.406138 + 0.913812i \(0.633125\pi\)
\(98\) 11.0000 1.11117
\(99\) 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.f.a.142.1 yes 2
5.2 odd 4 925.2.k.a.68.1 2
5.3 odd 4 185.2.k.b.68.1 yes 2
5.4 even 2 925.2.f.b.882.1 2
37.6 odd 4 185.2.k.b.117.1 yes 2
185.43 even 4 inner 185.2.f.a.43.1 ✓ 2
185.117 even 4 925.2.f.b.43.1 2
185.154 odd 4 925.2.k.a.857.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
185.2.f.a.43.1 ✓ 2 185.43 even 4 inner
185.2.f.a.142.1 yes 2 1.1 even 1 trivial
185.2.k.b.68.1 yes 2 5.3 odd 4
185.2.k.b.117.1 yes 2 37.6 odd 4
925.2.f.b.43.1 2 185.117 even 4
925.2.f.b.882.1 2 5.4 even 2
925.2.k.a.68.1 2 5.2 odd 4
925.2.k.a.857.1 2 185.154 odd 4