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Properties

Label 185.2.a.e.1.2
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 1.2
Root \(2.10563\) 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)
 
\(f(q)\) \(=\) \(q-1.13359 q^{2} -1.10563 q^{3} -0.714970 q^{4} -1.00000 q^{5} +1.25333 q^{6} +2.46164 q^{7} +3.07767 q^{8} -1.77758 q^{9} +1.13359 q^{10} +1.71497 q^{11} +0.790492 q^{12} +6.49255 q^{13} -2.79049 q^{14} +1.10563 q^{15} -2.05888 q^{16} +3.32980 q^{17} +2.01505 q^{18} +0.734568 q^{19} +0.714970 q^{20} -2.72166 q^{21} -1.94408 q^{22} -2.08603 q^{23} -3.40276 q^{24} +1.00000 q^{25} -7.35990 q^{26} +5.28224 q^{27} -1.76000 q^{28} -4.21126 q^{29} -1.25333 q^{30} +7.46459 q^{31} -3.82141 q^{32} -1.89612 q^{33} -3.77463 q^{34} -2.46164 q^{35} +1.27092 q^{36} +1.00000 q^{37} -0.832700 q^{38} -7.17836 q^{39} -3.07767 q^{40} +1.71497 q^{41} +3.08525 q^{42} +1.81885 q^{43} -1.22615 q^{44} +1.77758 q^{45} +2.36471 q^{46} -0.882270 q^{47} +2.27636 q^{48} -0.940340 q^{49} -1.13359 q^{50} -3.68152 q^{51} -4.64198 q^{52} -7.03066 q^{53} -5.98790 q^{54} -1.71497 q^{55} +7.57610 q^{56} -0.812159 q^{57} +4.77385 q^{58} +0.387867 q^{59} -0.790492 q^{60} -11.8224 q^{61} -8.46180 q^{62} -4.37577 q^{63} +8.44967 q^{64} -6.49255 q^{65} +2.14943 q^{66} +12.1086 q^{67} -2.38070 q^{68} +2.30638 q^{69} +2.79049 q^{70} +13.7486 q^{71} -5.47081 q^{72} +16.6719 q^{73} -1.13359 q^{74} -1.10563 q^{75} -0.525194 q^{76} +4.22163 q^{77} +8.13733 q^{78} +8.23253 q^{79} +2.05888 q^{80} -0.507447 q^{81} -1.94408 q^{82} -4.80275 q^{83} +1.94590 q^{84} -3.32980 q^{85} -2.06183 q^{86} +4.65609 q^{87} +5.27811 q^{88} -1.52506 q^{89} -2.01505 q^{90} +15.9823 q^{91} +1.49145 q^{92} -8.25307 q^{93} +1.00013 q^{94} -0.734568 q^{95} +4.22506 q^{96} -18.1588 q^{97} +1.06596 q^{98} -3.04850 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q + 2 q^{2} + 3 q^{3} + 10 q^{4} - 5 q^{5} - 6 q^{6} + 11 q^{7} + 6 q^{8} + 6 q^{9} - 2 q^{10} - 5 q^{11} - 2 q^{12} + 4 q^{13} - 8 q^{14} - 3 q^{15} + 16 q^{16} + 2 q^{18} - 4 q^{19} - 10 q^{20} + 3 q^{21}+ \cdots - 10 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.13359 −0.801570 −0.400785 − 0.916172i \(-0.631263\pi\)
−0.400785 + 0.916172i \(0.631263\pi\)
\(3\) −1.10563 −0.638335 −0.319168 − 0.947698i \(-0.603403\pi\)
−0.319168 + 0.947698i \(0.603403\pi\)
\(4\) −0.714970 −0.357485
\(5\) −1.00000 −0.447214
\(6\) 1.25333 0.511671
\(7\) 2.46164 0.930412 0.465206 − 0.885203i \(-0.345980\pi\)
0.465206 + 0.885203i \(0.345980\pi\)
\(8\) 3.07767 1.08812
\(9\) −1.77758 −0.592528
\(10\) 1.13359 0.358473
\(11\) 1.71497 0.517083 0.258541 − 0.966000i \(-0.416758\pi\)
0.258541 + 0.966000i \(0.416758\pi\)
\(12\) 0.790492 0.228195
\(13\) 6.49255 1.80071 0.900355 − 0.435156i \(-0.143307\pi\)
0.900355 + 0.435156i \(0.143307\pi\)
\(14\) −2.79049 −0.745790
\(15\) 1.10563 0.285472
\(16\) −2.05888 −0.514719
\(17\) 3.32980 0.807594 0.403797 − 0.914849i \(-0.367690\pi\)
0.403797 + 0.914849i \(0.367690\pi\)
\(18\) 2.01505 0.474953
\(19\) 0.734568 0.168521 0.0842607 − 0.996444i \(-0.473147\pi\)
0.0842607 + 0.996444i \(0.473147\pi\)
\(20\) 0.714970 0.159872
\(21\) −2.72166 −0.593915
\(22\) −1.94408 −0.414478
\(23\) −2.08603 −0.434968 −0.217484 − 0.976064i \(-0.569785\pi\)
−0.217484 + 0.976064i \(0.569785\pi\)
\(24\) −3.40276 −0.694585
\(25\) 1.00000 0.200000
\(26\) −7.35990 −1.44340
\(27\) 5.28224 1.01657
\(28\) −1.76000 −0.332608
\(29\) −4.21126 −0.782011 −0.391006 − 0.920388i \(-0.627873\pi\)
−0.391006 + 0.920388i \(0.627873\pi\)
\(30\) −1.25333 −0.228826
\(31\) 7.46459 1.34068 0.670340 − 0.742054i \(-0.266148\pi\)
0.670340 + 0.742054i \(0.266148\pi\)
\(32\) −3.82141 −0.675536
\(33\) −1.89612 −0.330072
\(34\) −3.77463 −0.647344
\(35\) −2.46164 −0.416093
\(36\) 1.27092 0.211820
\(37\) 1.00000 0.164399
\(38\) −0.832700 −0.135082
\(39\) −7.17836 −1.14946
\(40\) −3.07767 −0.486622
\(41\) 1.71497 0.267833 0.133917 − 0.990993i \(-0.457245\pi\)
0.133917 + 0.990993i \(0.457245\pi\)
\(42\) 3.08525 0.476064
\(43\) 1.81885 0.277372 0.138686 − 0.990336i \(-0.455712\pi\)
0.138686 + 0.990336i \(0.455712\pi\)
\(44\) −1.22615 −0.184849
\(45\) 1.77758 0.264986
\(46\) 2.36471 0.348657
\(47\) −0.882270 −0.128692 −0.0643462 − 0.997928i \(-0.520496\pi\)
−0.0643462 + 0.997928i \(0.520496\pi\)
\(48\) 2.27636 0.328564
\(49\) −0.940340 −0.134334
\(50\) −1.13359 −0.160314
\(51\) −3.68152 −0.515516
\(52\) −4.64198 −0.643727
\(53\) −7.03066 −0.965735 −0.482867 − 0.875693i \(-0.660405\pi\)
−0.482867 + 0.875693i \(0.660405\pi\)
\(54\) −5.98790 −0.814850
\(55\) −1.71497 −0.231247
\(56\) 7.57610 1.01240
\(57\) −0.812159 −0.107573
\(58\) 4.77385 0.626837
\(59\) 0.387867 0.0504959 0.0252480 − 0.999681i \(-0.491962\pi\)
0.0252480 + 0.999681i \(0.491962\pi\)
\(60\) −0.790492 −0.102052
\(61\) −11.8224 −1.51370 −0.756848 − 0.653590i \(-0.773262\pi\)
−0.756848 + 0.653590i \(0.773262\pi\)
\(62\) −8.46180 −1.07465
\(63\) −4.37577 −0.551295
\(64\) 8.44967 1.05621
\(65\) −6.49255 −0.805302
\(66\) 2.14943 0.264576
\(67\) 12.1086 1.47930 0.739649 − 0.672992i \(-0.234991\pi\)
0.739649 + 0.672992i \(0.234991\pi\)
\(68\) −2.38070 −0.288703
\(69\) 2.30638 0.277655
\(70\) 2.79049 0.333528
\(71\) 13.7486 1.63166 0.815828 − 0.578295i \(-0.196282\pi\)
0.815828 + 0.578295i \(0.196282\pi\)
\(72\) −5.47081 −0.644741
\(73\) 16.6719 1.95129 0.975646 − 0.219349i \(-0.0703933\pi\)
0.975646 + 0.219349i \(0.0703933\pi\)
\(74\) −1.13359 −0.131777
\(75\) −1.10563 −0.127667
\(76\) −0.525194 −0.0602439
\(77\) 4.22163 0.481100
\(78\) 8.13733 0.921371
\(79\) 8.23253 0.926232 0.463116 − 0.886298i \(-0.346731\pi\)
0.463116 + 0.886298i \(0.346731\pi\)
\(80\) 2.05888 0.230190
\(81\) −0.507447 −0.0563829
\(82\) −1.94408 −0.214687
\(83\) −4.80275 −0.527171 −0.263585 − 0.964636i \(-0.584905\pi\)
−0.263585 + 0.964636i \(0.584905\pi\)
\(84\) 1.94590 0.212316
\(85\) −3.32980 −0.361167
\(86\) −2.06183 −0.222333
\(87\) 4.65609 0.499185
\(88\) 5.27811 0.562648
\(89\) −1.52506 −0.161656 −0.0808280 − 0.996728i \(-0.525756\pi\)
−0.0808280 + 0.996728i \(0.525756\pi\)
\(90\) −2.01505 −0.212405
\(91\) 15.9823 1.67540
\(92\) 1.49145 0.155494
\(93\) −8.25307 −0.855804
\(94\) 1.00013 0.103156
\(95\) −0.734568 −0.0753650
\(96\) 4.22506 0.431218
\(97\) −18.1588 −1.84375 −0.921875 − 0.387487i \(-0.873343\pi\)
−0.921875 + 0.387487i \(0.873343\pi\)
\(98\) 1.06596 0.107678
\(99\) −3.04850 −0.306386
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.1.2 ✓ 5
3.2 odd 2 1665.2.a.p.1.4 5
4.3 odd 2 2960.2.a.w.1.4 5
5.2 odd 4 925.2.b.f.149.4 10
5.3 odd 4 925.2.b.f.149.7 10
5.4 even 2 925.2.a.f.1.4 5
7.6 odd 2 9065.2.a.k.1.2 5
15.14 odd 2 8325.2.a.ch.1.2 5
37.36 even 2 6845.2.a.f.1.4 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
185.2.a.e.1.2 ✓ 5 1.1 even 1 trivial
925.2.a.f.1.4 5 5.4 even 2
925.2.b.f.149.4 10 5.2 odd 4
925.2.b.f.149.7 10 5.3 odd 4
1665.2.a.p.1.4 5 3.2 odd 2
2960.2.a.w.1.4 5 4.3 odd 2
6845.2.a.f.1.4 5 37.36 even 2
8325.2.a.ch.1.2 5 15.14 odd 2
9065.2.a.k.1.2 5 7.6 odd 2