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Properties

Label 185.2.k.a.117.1
Level $185$
Weight $2$
Character 185.117
Analytic conductor $1.477$
Analytic rank $1$
Dimension $2$
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] = [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,-4] 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: \(1\)
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: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 117.1
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 185.117
Dual form 185.2.k.a.68.1

$q$-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)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +(-2.00000 - 2.00000i) q^{3} -1.00000 q^{4} +(-1.00000 - 2.00000i) q^{5} +(2.00000 + 2.00000i) q^{6} +3.00000 q^{8} +5.00000i q^{9} +(1.00000 + 2.00000i) q^{10} +4.00000i q^{11} +(2.00000 + 2.00000i) q^{12} -4.00000 q^{13} +(-2.00000 + 6.00000i) q^{15} -1.00000 q^{16} -2.00000i q^{17} -5.00000i q^{18} +(1.00000 + 2.00000i) q^{20} -4.00000i q^{22} +4.00000 q^{23} +(-6.00000 - 6.00000i) q^{24} +(-3.00000 + 4.00000i) q^{25} +4.00000 q^{26} +(4.00000 - 4.00000i) q^{27} +(-1.00000 - 1.00000i) q^{29} +(2.00000 - 6.00000i) q^{30} +(-6.00000 + 6.00000i) q^{31} -5.00000 q^{32} +(8.00000 - 8.00000i) q^{33} +2.00000i q^{34} -5.00000i q^{36} +(-1.00000 - 6.00000i) q^{37} +(8.00000 + 8.00000i) q^{39} +(-3.00000 - 6.00000i) q^{40} -12.0000 q^{43} -4.00000i q^{44} +(10.0000 - 5.00000i) q^{45} -4.00000 q^{46} +(8.00000 + 8.00000i) q^{47} +(2.00000 + 2.00000i) q^{48} -7.00000i q^{49} +(3.00000 - 4.00000i) q^{50} +(-4.00000 + 4.00000i) q^{51} +4.00000 q^{52} +(-9.00000 + 9.00000i) q^{53} +(-4.00000 + 4.00000i) q^{54} +(8.00000 - 4.00000i) q^{55} +(1.00000 + 1.00000i) q^{58} +(4.00000 - 4.00000i) q^{59} +(2.00000 - 6.00000i) q^{60} +(1.00000 - 1.00000i) q^{61} +(6.00000 - 6.00000i) q^{62} +7.00000 q^{64} +(4.00000 + 8.00000i) q^{65} +(-8.00000 + 8.00000i) q^{66} +(-6.00000 + 6.00000i) q^{67} +2.00000i q^{68} +(-8.00000 - 8.00000i) q^{69} -4.00000 q^{71} +15.0000i q^{72} +(-11.0000 - 11.0000i) q^{73} +(1.00000 + 6.00000i) q^{74} +(14.0000 - 2.00000i) q^{75} +(-8.00000 - 8.00000i) q^{78} +(-6.00000 + 6.00000i) q^{79} +(1.00000 + 2.00000i) q^{80} -1.00000 q^{81} +(-2.00000 + 2.00000i) q^{83} +(-4.00000 + 2.00000i) q^{85} +12.0000 q^{86} +4.00000i q^{87} +12.0000i q^{88} +(-1.00000 - 1.00000i) q^{89} +(-10.0000 + 5.00000i) q^{90} -4.00000 q^{92} +24.0000 q^{93} +(-8.00000 - 8.00000i) q^{94} +(10.0000 + 10.0000i) q^{96} +4.00000i q^{97} +7.00000i q^{98} -20.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} - 4 q^{3} - 2 q^{4} - 2 q^{5} + 4 q^{6} + 6 q^{8} + 2 q^{10} + 4 q^{12} - 8 q^{13} - 4 q^{15} - 2 q^{16} + 2 q^{20} + 8 q^{23} - 12 q^{24} - 6 q^{25} + 8 q^{26} + 8 q^{27} - 2 q^{29} + 4 q^{30}+ \cdots - 40 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107 −0.353553 − 0.935414i \(-0.615027\pi\)
−0.353553 + 0.935414i \(0.615027\pi\)
\(3\) −2.00000 − 2.00000i −1.15470 − 1.15470i −0.985599 − 0.169102i \(-0.945913\pi\)
−0.169102 − 0.985599i \(-0.554087\pi\)
\(4\) −1.00000 −0.500000
\(5\) −1.00000 − 2.00000i −0.447214 − 0.894427i
\(6\) 2.00000 + 2.00000i 0.816497 + 0.816497i
\(7\) 0 0 0.707107 − 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(8\) 3.00000 1.06066
\(9\) 5.00000i 1.66667i
\(10\) 1.00000 + 2.00000i 0.316228 + 0.632456i
\(11\) 4.00000i 1.20605i 0.797724 + 0.603023i \(0.206037\pi\)
−0.797724 + 0.603023i \(0.793963\pi\)
\(12\) 2.00000 + 2.00000i 0.577350 + 0.577350i
\(13\) −4.00000 −1.10940 −0.554700 − 0.832050i \(-0.687167\pi\)
−0.554700 + 0.832050i \(0.687167\pi\)
\(14\) 0 0
\(15\) −2.00000 + 6.00000i −0.516398 + 1.54919i
\(16\) −1.00000 −0.250000
\(17\) − 2.00000i − 0.485071i −0.970143 − 0.242536i \(-0.922021\pi\)
0.970143 − 0.242536i \(-0.0779791\pi\)
\(18\) − 5.00000i − 1.17851i
\(19\) 0 0 −0.707107 − 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(20\) 1.00000 + 2.00000i 0.223607 + 0.447214i
\(21\) 0 0
\(22\) − 4.00000i − 0.852803i
\(23\) 4.00000 0.834058 0.417029 − 0.908893i \(-0.363071\pi\)
0.417029 + 0.908893i \(0.363071\pi\)
\(24\) −6.00000 − 6.00000i −1.22474 − 1.22474i
\(25\) −3.00000 + 4.00000i −0.600000 + 0.800000i
\(26\) 4.00000 0.784465
\(27\) 4.00000 − 4.00000i 0.769800 − 0.769800i
\(28\) 0 0
\(29\) −1.00000 − 1.00000i −0.185695 − 0.185695i 0.608137 − 0.793832i \(-0.291917\pi\)
−0.793832 + 0.608137i \(0.791917\pi\)
\(30\) 2.00000 − 6.00000i 0.365148 − 1.09545i
\(31\) −6.00000 + 6.00000i −1.07763 + 1.07763i −0.0809104 + 0.996721i \(0.525783\pi\)
−0.996721 + 0.0809104i \(0.974217\pi\)
\(32\) −5.00000 −0.883883
\(33\) 8.00000 − 8.00000i 1.39262 − 1.39262i
\(34\) 2.00000i 0.342997i
\(35\) 0 0
\(36\) − 5.00000i − 0.833333i
\(37\) −1.00000 − 6.00000i −0.164399 − 0.986394i
\(38\) 0 0
\(39\) 8.00000 + 8.00000i 1.28103 + 1.28103i
\(40\) −3.00000 − 6.00000i −0.474342 − 0.948683i
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) 0 0
\(43\) −12.0000 −1.82998 −0.914991 − 0.403473i \(-0.867803\pi\)
−0.914991 + 0.403473i \(0.867803\pi\)
\(44\) − 4.00000i − 0.603023i
\(45\) 10.0000 − 5.00000i 1.49071 − 0.745356i
\(46\) −4.00000 −0.589768
\(47\) 8.00000 + 8.00000i 1.16692 + 1.16692i 0.982928 + 0.183992i \(0.0589021\pi\)
0.183992 + 0.982928i \(0.441098\pi\)
\(48\) 2.00000 + 2.00000i 0.288675 + 0.288675i
\(49\) − 7.00000i − 1.00000i
\(50\) 3.00000 − 4.00000i 0.424264 − 0.565685i
\(51\) −4.00000 + 4.00000i −0.560112 + 0.560112i
\(52\) 4.00000 0.554700
\(53\) −9.00000 + 9.00000i −1.23625 + 1.23625i −0.274721 + 0.961524i \(0.588586\pi\)
−0.961524 + 0.274721i \(0.911414\pi\)
\(54\) −4.00000 + 4.00000i −0.544331 + 0.544331i
\(55\) 8.00000 − 4.00000i 1.07872 − 0.539360i
\(56\) 0 0
\(57\) 0 0
\(58\) 1.00000 + 1.00000i 0.131306 + 0.131306i
\(59\) 4.00000 − 4.00000i 0.520756 − 0.520756i −0.397044 − 0.917800i \(-0.629964\pi\)
0.917800 + 0.397044i \(0.129964\pi\)
\(60\) 2.00000 − 6.00000i 0.258199 − 0.774597i
\(61\) 1.00000 − 1.00000i 0.128037 − 0.128037i −0.640184 − 0.768221i \(-0.721142\pi\)
0.768221 + 0.640184i \(0.221142\pi\)
\(62\) 6.00000 − 6.00000i 0.762001 − 0.762001i
\(63\) 0 0
\(64\) 7.00000 0.875000
\(65\) 4.00000 + 8.00000i 0.496139 + 0.992278i
\(66\) −8.00000 + 8.00000i −0.984732 + 0.984732i
\(67\) −6.00000 + 6.00000i −0.733017 + 0.733017i −0.971216 − 0.238200i \(-0.923443\pi\)
0.238200 + 0.971216i \(0.423443\pi\)
\(68\) 2.00000i 0.242536i
\(69\) −8.00000 − 8.00000i −0.963087 − 0.963087i
\(70\) 0 0
\(71\) −4.00000 −0.474713 −0.237356 − 0.971423i \(-0.576281\pi\)
−0.237356 + 0.971423i \(0.576281\pi\)
\(72\) 15.0000i 1.76777i
\(73\) −11.0000 − 11.0000i −1.28745 − 1.28745i −0.936329 − 0.351123i \(-0.885800\pi\)
−0.351123 − 0.936329i \(-0.614200\pi\)
\(74\) 1.00000 + 6.00000i 0.116248 + 0.697486i
\(75\) 14.0000 − 2.00000i 1.61658 − 0.230940i
\(76\) 0 0
\(77\) 0 0
\(78\) −8.00000 − 8.00000i −0.905822 − 0.905822i
\(79\) −6.00000 + 6.00000i −0.675053 + 0.675053i −0.958876 − 0.283824i \(-0.908397\pi\)
0.283824 + 0.958876i \(0.408397\pi\)
\(80\) 1.00000 + 2.00000i 0.111803 + 0.223607i
\(81\) −1.00000 −0.111111
\(82\) 0 0
\(83\) −2.00000 + 2.00000i −0.219529 + 0.219529i −0.808300 − 0.588771i \(-0.799612\pi\)
0.588771 + 0.808300i \(0.299612\pi\)
\(84\) 0 0
\(85\) −4.00000 + 2.00000i −0.433861 + 0.216930i
\(86\) 12.0000 1.29399
\(87\) 4.00000i 0.428845i
\(88\) 12.0000i 1.27920i
\(89\) −1.00000 − 1.00000i −0.106000 − 0.106000i 0.652118 − 0.758118i \(-0.273881\pi\)
−0.758118 + 0.652118i \(0.773881\pi\)
\(90\) −10.0000 + 5.00000i −1.05409 + 0.527046i
\(91\) 0 0
\(92\) −4.00000 −0.417029
\(93\) 24.0000 2.48868
\(94\) −8.00000 − 8.00000i −0.825137 − 0.825137i
\(95\) 0 0
\(96\) 10.0000 + 10.0000i 1.02062 + 1.02062i
\(97\) 4.00000i 0.406138i 0.979164 + 0.203069i \(0.0650917\pi\)
−0.979164 + 0.203069i \(0.934908\pi\)
\(98\) 7.00000i 0.707107i
\(99\) −20.0000 −2.01008
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.a.117.1 yes 2
5.2 odd 4 925.2.f.a.43.1 2
5.3 odd 4 185.2.f.b.43.1 ✓ 2
5.4 even 2 925.2.k.b.857.1 2
37.31 odd 4 185.2.f.b.142.1 yes 2
185.68 even 4 inner 185.2.k.a.68.1 yes 2
185.142 even 4 925.2.k.b.68.1 2
185.179 odd 4 925.2.f.a.882.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
185.2.f.b.43.1 ✓ 2 5.3 odd 4
185.2.f.b.142.1 yes 2 37.31 odd 4
185.2.k.a.68.1 yes 2 185.68 even 4 inner
185.2.k.a.117.1 yes 2 1.1 even 1 trivial
925.2.f.a.43.1 2 5.2 odd 4
925.2.f.a.882.1 2 185.179 odd 4
925.2.k.b.68.1 2 185.142 even 4
925.2.k.b.857.1 2 5.4 even 2