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Properties

Label 185.2.a.e.1.1
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 1.1
Root \(-1.38679\) 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)
 
\(f(q)\) \(=\) \(q-2.47408 q^{2} +2.38679 q^{3} +4.12105 q^{4} -1.00000 q^{5} -5.90509 q^{6} +4.78404 q^{7} -5.24765 q^{8} +2.69675 q^{9} +2.47408 q^{10} -3.12105 q^{11} +9.83607 q^{12} -2.81780 q^{13} -11.8361 q^{14} -2.38679 q^{15} +4.74097 q^{16} +6.37246 q^{17} -6.67196 q^{18} +0.114347 q^{19} -4.12105 q^{20} +11.4185 q^{21} +7.72172 q^{22} +5.62219 q^{23} -12.5250 q^{24} +1.00000 q^{25} +6.97145 q^{26} -0.723803 q^{27} +19.7153 q^{28} +2.77357 q^{29} +5.90509 q^{30} -6.67866 q^{31} -1.23424 q^{32} -7.44929 q^{33} -15.7660 q^{34} -4.78404 q^{35} +11.1134 q^{36} +1.00000 q^{37} -0.282904 q^{38} -6.72548 q^{39} +5.24765 q^{40} -3.12105 q^{41} -28.2502 q^{42} -8.57034 q^{43} -12.8620 q^{44} -2.69675 q^{45} -13.9097 q^{46} +3.40396 q^{47} +11.3157 q^{48} +15.8870 q^{49} -2.47408 q^{50} +15.2097 q^{51} -11.6123 q^{52} -10.2438 q^{53} +1.79074 q^{54} +3.12105 q^{55} -25.1049 q^{56} +0.272922 q^{57} -6.86203 q^{58} -9.11059 q^{59} -9.83607 q^{60} -5.55466 q^{61} +16.5235 q^{62} +12.9013 q^{63} -6.42836 q^{64} +2.81780 q^{65} +18.4301 q^{66} -7.84948 q^{67} +26.2612 q^{68} +13.4190 q^{69} +11.8361 q^{70} -4.33996 q^{71} -14.1516 q^{72} +3.22811 q^{73} -2.47408 q^{74} +2.38679 q^{75} +0.471231 q^{76} -14.9312 q^{77} +16.6394 q^{78} +15.3847 q^{79} -4.74097 q^{80} -9.81780 q^{81} +7.72172 q^{82} +5.68074 q^{83} +47.0561 q^{84} -6.37246 q^{85} +21.2037 q^{86} +6.61992 q^{87} +16.3782 q^{88} -9.95042 q^{89} +6.67196 q^{90} -13.4805 q^{91} +23.1693 q^{92} -15.9405 q^{93} -8.42165 q^{94} -0.114347 q^{95} -2.94586 q^{96} -5.62970 q^{97} -39.3057 q^{98} -8.41669 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q + 2 q^{2} + 3 q^{3} + 10 q^{4} - 5 q^{5} - 6 q^{6} + 11 q^{7} + 6 q^{8} + 6 q^{9} - 2 q^{10} - 5 q^{11} - 2 q^{12} + 4 q^{13} - 8 q^{14} - 3 q^{15} + 16 q^{16} + 2 q^{18} - 4 q^{19} - 10 q^{20} + 3 q^{21}+ \cdots - 10 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.47408 −1.74944 −0.874718 − 0.484632i \(-0.838953\pi\)
−0.874718 + 0.484632i \(0.838953\pi\)
\(3\) 2.38679 1.37801 0.689006 − 0.724756i \(-0.258047\pi\)
0.689006 + 0.724756i \(0.258047\pi\)
\(4\) 4.12105 2.06053
\(5\) −1.00000 −0.447214
\(6\) −5.90509 −2.41074
\(7\) 4.78404 1.80820 0.904098 − 0.427325i \(-0.140544\pi\)
0.904098 + 0.427325i \(0.140544\pi\)
\(8\) −5.24765 −1.85532
\(9\) 2.69675 0.898915
\(10\) 2.47408 0.782372
\(11\) −3.12105 −0.941033 −0.470516 − 0.882391i \(-0.655932\pi\)
−0.470516 + 0.882391i \(0.655932\pi\)
\(12\) 9.83607 2.83943
\(13\) −2.81780 −0.781517 −0.390758 − 0.920493i \(-0.627787\pi\)
−0.390758 + 0.920493i \(0.627787\pi\)
\(14\) −11.8361 −3.16332
\(15\) −2.38679 −0.616265
\(16\) 4.74097 1.18524
\(17\) 6.37246 1.54555 0.772774 − 0.634681i \(-0.218868\pi\)
0.772774 + 0.634681i \(0.218868\pi\)
\(18\) −6.67196 −1.57259
\(19\) 0.114347 0.0262330 0.0131165 − 0.999914i \(-0.495825\pi\)
0.0131165 + 0.999914i \(0.495825\pi\)
\(20\) −4.12105 −0.921496
\(21\) 11.4185 2.49171
\(22\) 7.72172 1.64628
\(23\) 5.62219 1.17231 0.586153 − 0.810200i \(-0.300642\pi\)
0.586153 + 0.810200i \(0.300642\pi\)
\(24\) −12.5250 −2.55666
\(25\) 1.00000 0.200000
\(26\) 6.97145 1.36721
\(27\) −0.723803 −0.139296
\(28\) 19.7153 3.72584
\(29\) 2.77357 0.515039 0.257520 − 0.966273i \(-0.417095\pi\)
0.257520 + 0.966273i \(0.417095\pi\)
\(30\) 5.90509 1.07812
\(31\) −6.67866 −1.19952 −0.599761 − 0.800179i \(-0.704738\pi\)
−0.599761 + 0.800179i \(0.704738\pi\)
\(32\) −1.23424 −0.218184
\(33\) −7.44929 −1.29675
\(34\) −15.7660 −2.70384
\(35\) −4.78404 −0.808650
\(36\) 11.1134 1.85224
\(37\) 1.00000 0.164399
\(38\) −0.282904 −0.0458930
\(39\) −6.72548 −1.07694
\(40\) 5.24765 0.829726
\(41\) −3.12105 −0.487427 −0.243713 − 0.969847i \(-0.578366\pi\)
−0.243713 + 0.969847i \(0.578366\pi\)
\(42\) −28.2502 −4.35910
\(43\) −8.57034 −1.30696 −0.653482 − 0.756942i \(-0.726693\pi\)
−0.653482 + 0.756942i \(0.726693\pi\)
\(44\) −12.8620 −1.93902
\(45\) −2.69675 −0.402007
\(46\) −13.9097 −2.05088
\(47\) 3.40396 0.496518 0.248259 − 0.968694i \(-0.420142\pi\)
0.248259 + 0.968694i \(0.420142\pi\)
\(48\) 11.3157 1.63328
\(49\) 15.8870 2.26957
\(50\) −2.47408 −0.349887
\(51\) 15.2097 2.12978
\(52\) −11.6123 −1.61034
\(53\) −10.2438 −1.40709 −0.703546 − 0.710650i \(-0.748401\pi\)
−0.703546 + 0.710650i \(0.748401\pi\)
\(54\) 1.79074 0.243689
\(55\) 3.12105 0.420843
\(56\) −25.1049 −3.35479
\(57\) 0.272922 0.0361494
\(58\) −6.86203 −0.901028
\(59\) −9.11059 −1.18610 −0.593049 − 0.805167i \(-0.702076\pi\)
−0.593049 + 0.805167i \(0.702076\pi\)
\(60\) −9.83607 −1.26983
\(61\) −5.55466 −0.711201 −0.355601 − 0.934638i \(-0.615724\pi\)
−0.355601 + 0.934638i \(0.615724\pi\)
\(62\) 16.5235 2.09849
\(63\) 12.9013 1.62541
\(64\) −6.42836 −0.803544
\(65\) 2.81780 0.349505
\(66\) 18.4301 2.26859
\(67\) −7.84948 −0.958967 −0.479484 − 0.877551i \(-0.659176\pi\)
−0.479484 + 0.877551i \(0.659176\pi\)
\(68\) 26.2612 3.18464
\(69\) 13.4190 1.61545
\(70\) 11.8361 1.41468
\(71\) −4.33996 −0.515059 −0.257529 − 0.966270i \(-0.582908\pi\)
−0.257529 + 0.966270i \(0.582908\pi\)
\(72\) −14.1516 −1.66778
\(73\) 3.22811 0.377822 0.188911 − 0.981994i \(-0.439504\pi\)
0.188911 + 0.981994i \(0.439504\pi\)
\(74\) −2.47408 −0.287606
\(75\) 2.38679 0.275602
\(76\) 0.471231 0.0540539
\(77\) −14.9312 −1.70157
\(78\) 16.6394 1.88404
\(79\) 15.3847 1.73091 0.865457 − 0.500983i \(-0.167028\pi\)
0.865457 + 0.500983i \(0.167028\pi\)
\(80\) −4.74097 −0.530057
\(81\) −9.81780 −1.09087
\(82\) 7.72172 0.852722
\(83\) 5.68074 0.623542 0.311771 − 0.950157i \(-0.399078\pi\)
0.311771 + 0.950157i \(0.399078\pi\)
\(84\) 47.0561 5.13424
\(85\) −6.37246 −0.691190
\(86\) 21.2037 2.28645
\(87\) 6.61992 0.709730
\(88\) 16.3782 1.74592
\(89\) −9.95042 −1.05474 −0.527371 − 0.849635i \(-0.676822\pi\)
−0.527371 + 0.849635i \(0.676822\pi\)
\(90\) 6.67196 0.703286
\(91\) −13.4805 −1.41314
\(92\) 23.1693 2.41557
\(93\) −15.9405 −1.65296
\(94\) −8.42165 −0.868627
\(95\) −0.114347 −0.0117318
\(96\) −2.94586 −0.300660
\(97\) −5.62970 −0.571610 −0.285805 − 0.958288i \(-0.592261\pi\)
−0.285805 + 0.958288i \(0.592261\pi\)
\(98\) −39.3057 −3.97047
\(99\) −8.41669 −0.845909
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.1.1 ✓ 5
3.2 odd 2 1665.2.a.p.1.5 5
4.3 odd 2 2960.2.a.w.1.2 5
5.2 odd 4 925.2.b.f.149.2 10
5.3 odd 4 925.2.b.f.149.9 10
5.4 even 2 925.2.a.f.1.5 5
7.6 odd 2 9065.2.a.k.1.1 5
15.14 odd 2 8325.2.a.ch.1.1 5
37.36 even 2 6845.2.a.f.1.5 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
185.2.a.e.1.1 ✓ 5 1.1 even 1 trivial
925.2.a.f.1.5 5 5.4 even 2
925.2.b.f.149.2 10 5.2 odd 4
925.2.b.f.149.9 10 5.3 odd 4
1665.2.a.p.1.5 5 3.2 odd 2
2960.2.a.w.1.2 5 4.3 odd 2
6845.2.a.f.1.5 5 37.36 even 2
8325.2.a.ch.1.1 5 15.14 odd 2
9065.2.a.k.1.1 5 7.6 odd 2