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Properties

Label 185.2.f.b.142.1
Level $185$
Weight $2$
Character 185.142
Analytic conductor $1.477$
Analytic rank $0$
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(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 = [2,0,4] 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: \(0\)
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.b.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)
 
\(f(q)\) \(=\) \(q-1.00000i q^{2} +(2.00000 + 2.00000i) q^{3} +1.00000 q^{4} +(-2.00000 + 1.00000i) q^{5} +(2.00000 - 2.00000i) q^{6} -3.00000i q^{8} +5.00000i q^{9} +(1.00000 + 2.00000i) q^{10} -4.00000i q^{11} +(2.00000 + 2.00000i) q^{12} +4.00000i q^{13} +(-6.00000 - 2.00000i) q^{15} -1.00000 q^{16} -2.00000 q^{17} +5.00000 q^{18} +(-2.00000 + 1.00000i) q^{20} -4.00000 q^{22} -4.00000i q^{23} +(6.00000 - 6.00000i) q^{24} +(3.00000 - 4.00000i) q^{25} +4.00000 q^{26} +(-4.00000 + 4.00000i) q^{27} +(1.00000 - 1.00000i) q^{29} +(-2.00000 + 6.00000i) q^{30} +(-6.00000 - 6.00000i) q^{31} -5.00000i q^{32} +(8.00000 - 8.00000i) q^{33} +2.00000i q^{34} +5.00000i q^{36} +(-6.00000 - 1.00000i) q^{37} +(-8.00000 + 8.00000i) q^{39} +(3.00000 + 6.00000i) q^{40} +12.0000i q^{43} -4.00000i q^{44} +(-5.00000 - 10.0000i) q^{45} -4.00000 q^{46} +(8.00000 + 8.00000i) q^{47} +(-2.00000 - 2.00000i) q^{48} -7.00000i q^{49} +(-4.00000 - 3.00000i) q^{50} +(-4.00000 - 4.00000i) q^{51} +4.00000i q^{52} +(-9.00000 + 9.00000i) q^{53} +(4.00000 + 4.00000i) q^{54} +(4.00000 + 8.00000i) q^{55} +(-1.00000 - 1.00000i) q^{58} +(-4.00000 - 4.00000i) q^{59} +(-6.00000 - 2.00000i) q^{60} +(1.00000 + 1.00000i) q^{61} +(-6.00000 + 6.00000i) q^{62} -7.00000 q^{64} +(-4.00000 - 8.00000i) q^{65} +(-8.00000 - 8.00000i) q^{66} +(6.00000 - 6.00000i) q^{67} -2.00000 q^{68} +(8.00000 - 8.00000i) q^{69} -4.00000 q^{71} +15.0000 q^{72} +(11.0000 + 11.0000i) q^{73} +(-1.00000 + 6.00000i) q^{74} +(14.0000 - 2.00000i) q^{75} +(8.00000 + 8.00000i) q^{78} +(6.00000 + 6.00000i) q^{79} +(2.00000 - 1.00000i) q^{80} -1.00000 q^{81} +(-2.00000 + 2.00000i) q^{83} +(4.00000 - 2.00000i) q^{85} +12.0000 q^{86} +4.00000 q^{87} -12.0000 q^{88} +(1.00000 - 1.00000i) q^{89} +(-10.0000 + 5.00000i) q^{90} -4.00000i q^{92} -24.0000i q^{93} +(8.00000 - 8.00000i) q^{94} +(10.0000 - 10.0000i) q^{96} +4.00000 q^{97} -7.00000 q^{98} +20.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 4 q^{3} + 2 q^{4} - 4 q^{5} + 4 q^{6} + 2 q^{10} + 4 q^{12} - 12 q^{15} - 2 q^{16} - 4 q^{17} + 10 q^{18} - 4 q^{20} - 8 q^{22} + 12 q^{24} + 6 q^{25} + 8 q^{26} - 8 q^{27} + 2 q^{29} - 4 q^{30} - 12 q^{31}+ \cdots + 40 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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 \(-0.115027\pi\)
\(3\) 2.00000 + 2.00000i 1.15470 + 1.15470i 0.985599 + 0.169102i \(0.0540867\pi\)
0.169102 + 0.985599i \(0.445913\pi\)
\(4\) 1.00000 0.500000
\(5\) −2.00000 + 1.00000i −0.894427 + 0.447214i
\(6\) 2.00000 − 2.00000i 0.816497 − 0.816497i
\(7\) 0 0 0.707107 − 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(8\) − 3.00000i − 1.06066i
\(9\) 5.00000i 1.66667i
\(10\) 1.00000 + 2.00000i 0.316228 + 0.632456i
\(11\) − 4.00000i − 1.20605i −0.797724 − 0.603023i \(-0.793963\pi\)
0.797724 − 0.603023i \(-0.206037\pi\)
\(12\) 2.00000 + 2.00000i 0.577350 + 0.577350i
\(13\) 4.00000i 1.10940i 0.832050 + 0.554700i \(0.187167\pi\)
−0.832050 + 0.554700i \(0.812833\pi\)
\(14\) 0 0
\(15\) −6.00000 − 2.00000i −1.54919 − 0.516398i
\(16\) −1.00000 −0.250000
\(17\) −2.00000 −0.485071 −0.242536 − 0.970143i \(-0.577979\pi\)
−0.242536 + 0.970143i \(0.577979\pi\)
\(18\) 5.00000 1.17851
\(19\) 0 0 0.707107 − 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(20\) −2.00000 + 1.00000i −0.447214 + 0.223607i
\(21\) 0 0
\(22\) −4.00000 −0.852803
\(23\) − 4.00000i − 0.834058i −0.908893 − 0.417029i \(-0.863071\pi\)
0.908893 − 0.417029i \(-0.136929\pi\)
\(24\) 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 \(-0.708083\pi\)
0.793832 + 0.608137i \(0.208083\pi\)
\(30\) −2.00000 + 6.00000i −0.365148 + 1.09545i
\(31\) −6.00000 − 6.00000i −1.07763 − 1.07763i −0.996721 − 0.0809104i \(-0.974217\pi\)
−0.0809104 − 0.996721i \(-0.525783\pi\)
\(32\) − 5.00000i − 0.883883i
\(33\) 8.00000 − 8.00000i 1.39262 − 1.39262i
\(34\) 2.00000i 0.342997i
\(35\) 0 0
\(36\) 5.00000i 0.833333i
\(37\) −6.00000 − 1.00000i −0.986394 − 0.164399i
\(38\) 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
\(43\) 12.0000i 1.82998i 0.403473 + 0.914991i \(0.367803\pi\)
−0.403473 + 0.914991i \(0.632197\pi\)
\(44\) − 4.00000i − 0.603023i
\(45\) −5.00000 − 10.0000i −0.745356 − 1.49071i
\(46\) −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
\(50\) −4.00000 − 3.00000i −0.565685 − 0.424264i
\(51\) −4.00000 − 4.00000i −0.560112 − 0.560112i
\(52\) 4.00000i 0.554700i
\(53\) −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
\(55\) 4.00000 + 8.00000i 0.539360 + 1.07872i
\(56\) 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 \(-0.370036\pi\)
−0.917800 + 0.397044i \(0.870036\pi\)
\(60\) −6.00000 − 2.00000i −0.774597 − 0.258199i
\(61\) 1.00000 + 1.00000i 0.128037 + 0.128037i 0.768221 − 0.640184i \(-0.221142\pi\)
−0.640184 + 0.768221i \(0.721142\pi\)
\(62\) −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 −0.238200 − 0.971216i \(-0.576557\pi\)
0.971216 + 0.238200i \(0.0765572\pi\)
\(68\) −2.00000 −0.242536
\(69\) 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 \(0.576281\pi\)
\(72\) 15.0000 1.76777
\(73\) 11.0000 + 11.0000i 1.28745 + 1.28745i 0.936329 + 0.351123i \(0.114200\pi\)
0.351123 + 0.936329i \(0.385800\pi\)
\(74\) −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 \(-0.0916031\pi\)
−0.283824 + 0.958876i \(0.591603\pi\)
\(80\) 2.00000 − 1.00000i 0.223607 − 0.111803i
\(81\) −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
\(87\) 4.00000 0.428845
\(88\) −12.0000 −1.27920
\(89\) 1.00000 − 1.00000i 0.106000 − 0.106000i −0.652118 − 0.758118i \(-0.726119\pi\)
0.758118 + 0.652118i \(0.226119\pi\)
\(90\) −10.0000 + 5.00000i −1.05409 + 0.527046i
\(91\) 0 0
\(92\) − 4.00000i − 0.417029i
\(93\) − 24.0000i − 2.48868i
\(94\) 8.00000 − 8.00000i 0.825137 − 0.825137i
\(95\) 0 0
\(96\) 10.0000 − 10.0000i 1.02062 − 1.02062i
\(97\) 4.00000 0.406138 0.203069 − 0.979164i \(-0.434908\pi\)
0.203069 + 0.979164i \(0.434908\pi\)
\(98\) −7.00000 −0.707107
\(99\) 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.f.b.142.1 yes 2
5.2 odd 4 925.2.k.b.68.1 2
5.3 odd 4 185.2.k.a.68.1 yes 2
5.4 even 2 925.2.f.a.882.1 2
37.6 odd 4 185.2.k.a.117.1 yes 2
185.43 even 4 inner 185.2.f.b.43.1 ✓ 2
185.117 even 4 925.2.f.a.43.1 2
185.154 odd 4 925.2.k.b.857.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
185.2.f.b.43.1 ✓ 2 185.43 even 4 inner
185.2.f.b.142.1 yes 2 1.1 even 1 trivial
185.2.k.a.68.1 yes 2 5.3 odd 4
185.2.k.a.117.1 yes 2 37.6 odd 4
925.2.f.a.43.1 2 185.117 even 4
925.2.f.a.882.1 2 5.4 even 2
925.2.k.b.68.1 2 5.2 odd 4
925.2.k.b.857.1 2 185.154 odd 4