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Properties

Label 185.2.a.e.1.5
Level $185$
Weight $2$
Character 185.1
Self dual yes
Analytic conductor $1.477$
Analytic rank $0$
Dimension $5$
CM no
Inner twists $1$

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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(1,185)] 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("185.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(185, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 185 = 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 185.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [5,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(1.47723243739\)
Analytic rank: \(0\)
Dimension: \(5\)
Coefficient field: 5.5.973904.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - 2x^{4} - 8x^{3} + 6x^{2} + 19x + 6 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.5
Root \(3.29298\) of defining polynomial
Character \(\chi\) \(=\) 185.1

$q$-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)
 
\(f(q)\) \(=\) \(q+2.72362 q^{2} -2.29298 q^{3} +5.41809 q^{4} -1.00000 q^{5} -6.24519 q^{6} +3.82710 q^{7} +9.30957 q^{8} +2.25774 q^{9} -2.72362 q^{10} -4.41809 q^{11} -12.4236 q^{12} -3.67583 q^{13} +10.4236 q^{14} +2.29298 q^{15} +14.5195 q^{16} -2.28688 q^{17} +6.14922 q^{18} -2.39037 q^{19} -5.41809 q^{20} -8.77545 q^{21} -12.0332 q^{22} -0.265251 q^{23} -21.3466 q^{24} +1.00000 q^{25} -10.0116 q^{26} +1.70198 q^{27} +20.7356 q^{28} -6.58595 q^{29} +6.24519 q^{30} +2.34076 q^{31} +20.9265 q^{32} +10.1306 q^{33} -6.22860 q^{34} -3.82710 q^{35} +12.2326 q^{36} +1.00000 q^{37} -6.51044 q^{38} +8.42859 q^{39} -9.30957 q^{40} -4.41809 q^{41} -23.9010 q^{42} +7.71249 q^{43} -23.9376 q^{44} -2.25774 q^{45} -0.722443 q^{46} +10.9285 q^{47} -33.2929 q^{48} +7.64669 q^{49} +2.72362 q^{50} +5.24377 q^{51} -19.9160 q^{52} -0.109574 q^{53} +4.63555 q^{54} +4.41809 q^{55} +35.6286 q^{56} +5.48105 q^{57} -17.9376 q^{58} -2.00504 q^{59} +12.4236 q^{60} +3.96271 q^{61} +6.37534 q^{62} +8.64059 q^{63} +27.9567 q^{64} +3.67583 q^{65} +27.5918 q^{66} +6.80664 q^{67} -12.3905 q^{68} +0.608215 q^{69} -10.4236 q^{70} -5.79485 q^{71} +21.0186 q^{72} -0.140654 q^{73} +2.72362 q^{74} -2.29298 q^{75} -12.9512 q^{76} -16.9085 q^{77} +22.9563 q^{78} -6.62418 q^{79} -14.5195 q^{80} -10.6758 q^{81} -12.0332 q^{82} +13.9904 q^{83} -47.5462 q^{84} +2.28688 q^{85} +21.0059 q^{86} +15.1014 q^{87} -41.1305 q^{88} +14.8139 q^{89} -6.14922 q^{90} -14.0678 q^{91} -1.43716 q^{92} -5.36731 q^{93} +29.7651 q^{94} +2.39037 q^{95} -47.9839 q^{96} -8.94394 q^{97} +20.8266 q^{98} -9.97490 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q + 2 q^{2} + 3 q^{3} + 10 q^{4} - 5 q^{5} - 6 q^{6} + 11 q^{7} + 6 q^{8} + 6 q^{9} - 2 q^{10} - 5 q^{11} - 2 q^{12} + 4 q^{13} - 8 q^{14} - 3 q^{15} + 16 q^{16} + 2 q^{18} - 4 q^{19} - 10 q^{20} + 3 q^{21}+ \cdots - 10 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.72362 1.92589 0.962944 − 0.269701i \(-0.0869249\pi\)
0.962944 + 0.269701i \(0.0869249\pi\)
\(3\) −2.29298 −1.32385 −0.661925 − 0.749570i \(-0.730260\pi\)
−0.661925 + 0.749570i \(0.730260\pi\)
\(4\) 5.41809 2.70905
\(5\) −1.00000 −0.447214
\(6\) −6.24519 −2.54959
\(7\) 3.82710 1.44651 0.723254 − 0.690582i \(-0.242646\pi\)
0.723254 + 0.690582i \(0.242646\pi\)
\(8\) 9.30957 3.29143
\(9\) 2.25774 0.752580
\(10\) −2.72362 −0.861283
\(11\) −4.41809 −1.33210 −0.666052 − 0.745905i \(-0.732017\pi\)
−0.666052 + 0.745905i \(0.732017\pi\)
\(12\) −12.4236 −3.58637
\(13\) −3.67583 −1.01949 −0.509746 − 0.860325i \(-0.670261\pi\)
−0.509746 + 0.860325i \(0.670261\pi\)
\(14\) 10.4236 2.78581
\(15\) 2.29298 0.592044
\(16\) 14.5195 3.62988
\(17\) −2.28688 −0.554651 −0.277325 − 0.960776i \(-0.589448\pi\)
−0.277325 + 0.960776i \(0.589448\pi\)
\(18\) 6.14922 1.44938
\(19\) −2.39037 −0.548387 −0.274194 − 0.961674i \(-0.588411\pi\)
−0.274194 + 0.961674i \(0.588411\pi\)
\(20\) −5.41809 −1.21152
\(21\) −8.77545 −1.91496
\(22\) −12.0332 −2.56548
\(23\) −0.265251 −0.0553087 −0.0276544 − 0.999618i \(-0.508804\pi\)
−0.0276544 + 0.999618i \(0.508804\pi\)
\(24\) −21.3466 −4.35736
\(25\) 1.00000 0.200000
\(26\) −10.0116 −1.96343
\(27\) 1.70198 0.327547
\(28\) 20.7356 3.91865
\(29\) −6.58595 −1.22298 −0.611490 − 0.791252i \(-0.709430\pi\)
−0.611490 + 0.791252i \(0.709430\pi\)
\(30\) 6.24519 1.14021
\(31\) 2.34076 0.420413 0.210207 − 0.977657i \(-0.432586\pi\)
0.210207 + 0.977657i \(0.432586\pi\)
\(32\) 20.9265 3.69931
\(33\) 10.1306 1.76351
\(34\) −6.22860 −1.06820
\(35\) −3.82710 −0.646898
\(36\) 12.2326 2.03877
\(37\) 1.00000 0.164399
\(38\) −6.51044 −1.05613
\(39\) 8.42859 1.34965
\(40\) −9.30957 −1.47197
\(41\) −4.41809 −0.689990 −0.344995 − 0.938605i \(-0.612119\pi\)
−0.344995 + 0.938605i \(0.612119\pi\)
\(42\) −23.9010 −3.68800
\(43\) 7.71249 1.17614 0.588072 − 0.808809i \(-0.299887\pi\)
0.588072 + 0.808809i \(0.299887\pi\)
\(44\) −23.9376 −3.60873
\(45\) −2.25774 −0.336564
\(46\) −0.722443 −0.106518
\(47\) 10.9285 1.59409 0.797045 − 0.603920i \(-0.206395\pi\)
0.797045 + 0.603920i \(0.206395\pi\)
\(48\) −33.2929 −4.80542
\(49\) 7.64669 1.09238
\(50\) 2.72362 0.385178
\(51\) 5.24377 0.734275
\(52\) −19.9160 −2.76185
\(53\) −0.109574 −0.0150512 −0.00752559 − 0.999972i \(-0.502395\pi\)
−0.00752559 + 0.999972i \(0.502395\pi\)
\(54\) 4.63555 0.630819
\(55\) 4.41809 0.595735
\(56\) 35.6286 4.76108
\(57\) 5.48105 0.725983
\(58\) −17.9376 −2.35532
\(59\) −2.00504 −0.261034 −0.130517 − 0.991446i \(-0.541664\pi\)
−0.130517 + 0.991446i \(0.541664\pi\)
\(60\) 12.4236 1.60387
\(61\) 3.96271 0.507374 0.253687 − 0.967286i \(-0.418357\pi\)
0.253687 + 0.967286i \(0.418357\pi\)
\(62\) 6.37534 0.809669
\(63\) 8.64059 1.08861
\(64\) 27.9567 3.49458
\(65\) 3.67583 0.455931
\(66\) 27.5918 3.39632
\(67\) 6.80664 0.831563 0.415782 − 0.909464i \(-0.363508\pi\)
0.415782 + 0.909464i \(0.363508\pi\)
\(68\) −12.3905 −1.50257
\(69\) 0.608215 0.0732205
\(70\) −10.4236 −1.24585
\(71\) −5.79485 −0.687722 −0.343861 − 0.939020i \(-0.611735\pi\)
−0.343861 + 0.939020i \(0.611735\pi\)
\(72\) 21.0186 2.47706
\(73\) −0.140654 −0.0164623 −0.00823116 − 0.999966i \(-0.502620\pi\)
−0.00823116 + 0.999966i \(0.502620\pi\)
\(74\) 2.72362 0.316614
\(75\) −2.29298 −0.264770
\(76\) −12.9512 −1.48561
\(77\) −16.9085 −1.92690
\(78\) 22.9563 2.59928
\(79\) −6.62418 −0.745278 −0.372639 − 0.927976i \(-0.621547\pi\)
−0.372639 + 0.927976i \(0.621547\pi\)
\(80\) −14.5195 −1.62333
\(81\) −10.6758 −1.18620
\(82\) −12.0332 −1.32884
\(83\) 13.9904 1.53565 0.767825 − 0.640660i \(-0.221339\pi\)
0.767825 + 0.640660i \(0.221339\pi\)
\(84\) −47.5462 −5.18771
\(85\) 2.28688 0.248047
\(86\) 21.0059 2.26512
\(87\) 15.1014 1.61904
\(88\) −41.1305 −4.38453
\(89\) 14.8139 1.57027 0.785136 − 0.619323i \(-0.212593\pi\)
0.785136 + 0.619323i \(0.212593\pi\)
\(90\) −6.14922 −0.648185
\(91\) −14.0678 −1.47470
\(92\) −1.43716 −0.149834
\(93\) −5.36731 −0.556565
\(94\) 29.7651 3.07004
\(95\) 2.39037 0.245246
\(96\) −47.9839 −4.89734
\(97\) −8.94394 −0.908119 −0.454060 − 0.890971i \(-0.650025\pi\)
−0.454060 + 0.890971i \(0.650025\pi\)
\(98\) 20.8266 2.10381
\(99\) −9.97490 −1.00252
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.a.e.1.5 ✓ 5
3.2 odd 2 1665.2.a.p.1.1 5
4.3 odd 2 2960.2.a.w.1.5 5
5.2 odd 4 925.2.b.f.149.10 10
5.3 odd 4 925.2.b.f.149.1 10
5.4 even 2 925.2.a.f.1.1 5
7.6 odd 2 9065.2.a.k.1.5 5
15.14 odd 2 8325.2.a.ch.1.5 5
37.36 even 2 6845.2.a.f.1.1 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
185.2.a.e.1.5 ✓ 5 1.1 even 1 trivial
925.2.a.f.1.1 5 5.4 even 2
925.2.b.f.149.1 10 5.3 odd 4
925.2.b.f.149.10 10 5.2 odd 4
1665.2.a.p.1.1 5 3.2 odd 2
2960.2.a.w.1.5 5 4.3 odd 2
6845.2.a.f.1.1 5 37.36 even 2
8325.2.a.ch.1.5 5 15.14 odd 2
9065.2.a.k.1.5 5 7.6 odd 2