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Properties

Label 1850.2.a.bd.1.5
Level $1850$
Weight $2$
Character 1850.1
Self dual yes
Analytic conductor $14.772$
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] = [1850,2,Mod(1,1850)] 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("1850.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1850, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1850 = 2 \cdot 5^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1850.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [5,-5,0,5,0,0,1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(14.7723243739\)
Analytic rank: \(0\)
Dimension: \(5\)
Coefficient field: 5.5.1791440.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - 9x^{3} + 13x - 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 370)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.5
Root \(2.72987\) of defining polynomial
Character \(\chi\) \(=\) 1850.1

$q$-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)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +2.72987 q^{3} +1.00000 q^{4} -2.72987 q^{6} +4.14336 q^{7} -1.00000 q^{8} +4.45216 q^{9} -4.76853 q^{11} +2.72987 q^{12} +3.91744 q^{13} -4.14336 q^{14} +1.00000 q^{16} -3.31637 q^{17} -4.45216 q^{18} -1.85109 q^{19} +11.3108 q^{21} +4.76853 q^{22} +1.54229 q^{23} -2.72987 q^{24} -3.91744 q^{26} +3.96421 q^{27} +4.14336 q^{28} +8.87323 q^{29} +9.75286 q^{31} -1.00000 q^{32} -13.0174 q^{33} +3.31637 q^{34} +4.45216 q^{36} -1.00000 q^{37} +1.85109 q^{38} +10.6941 q^{39} -5.06977 q^{41} -11.3108 q^{42} +9.99446 q^{43} -4.76853 q^{44} -1.54229 q^{46} -4.82700 q^{47} +2.72987 q^{48} +10.1675 q^{49} -9.05324 q^{51} +3.91744 q^{52} +5.13611 q^{53} -3.96421 q^{54} -4.14336 q^{56} -5.05324 q^{57} -8.87323 q^{58} -1.05324 q^{59} -4.14422 q^{61} -9.75286 q^{62} +18.4469 q^{63} +1.00000 q^{64} +13.0174 q^{66} -1.00811 q^{67} -3.31637 q^{68} +4.21025 q^{69} +6.45248 q^{71} -4.45216 q^{72} -10.7714 q^{73} +1.00000 q^{74} -1.85109 q^{76} -19.7578 q^{77} -10.6941 q^{78} -1.19856 q^{79} -2.53473 q^{81} +5.06977 q^{82} -10.6932 q^{83} +11.3108 q^{84} -9.99446 q^{86} +24.2227 q^{87} +4.76853 q^{88} +7.29569 q^{89} +16.2314 q^{91} +1.54229 q^{92} +26.6240 q^{93} +4.82700 q^{94} -2.72987 q^{96} -14.8988 q^{97} -10.1675 q^{98} -21.2303 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q - 5 q^{2} + 5 q^{4} + q^{7} - 5 q^{8} + 3 q^{9} + 3 q^{11} + 6 q^{13} - q^{14} + 5 q^{16} - 9 q^{17} - 3 q^{18} + 4 q^{19} + 16 q^{21} - 3 q^{22} - 6 q^{23} - 6 q^{26} + q^{28} + 11 q^{29} + 23 q^{31}+ \cdots - 11 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) 2.72987 1.57609 0.788044 − 0.615619i \(-0.211094\pi\)
0.788044 + 0.615619i \(0.211094\pi\)
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) −2.72987 −1.11446
\(7\) 4.14336 1.56604 0.783022 − 0.621994i \(-0.213677\pi\)
0.783022 + 0.621994i \(0.213677\pi\)
\(8\) −1.00000 −0.353553
\(9\) 4.45216 1.48405
\(10\) 0 0
\(11\) −4.76853 −1.43777 −0.718883 − 0.695131i \(-0.755346\pi\)
−0.718883 + 0.695131i \(0.755346\pi\)
\(12\) 2.72987 0.788044
\(13\) 3.91744 1.08650 0.543251 − 0.839570i \(-0.317193\pi\)
0.543251 + 0.839570i \(0.317193\pi\)
\(14\) −4.14336 −1.10736
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −3.31637 −0.804337 −0.402169 − 0.915566i \(-0.631743\pi\)
−0.402169 + 0.915566i \(0.631743\pi\)
\(18\) −4.45216 −1.04939
\(19\) −1.85109 −0.424670 −0.212335 − 0.977197i \(-0.568107\pi\)
−0.212335 + 0.977197i \(0.568107\pi\)
\(20\) 0 0
\(21\) 11.3108 2.46822
\(22\) 4.76853 1.01665
\(23\) 1.54229 0.321590 0.160795 − 0.986988i \(-0.448594\pi\)
0.160795 + 0.986988i \(0.448594\pi\)
\(24\) −2.72987 −0.557231
\(25\) 0 0
\(26\) −3.91744 −0.768273
\(27\) 3.96421 0.762913
\(28\) 4.14336 0.783022
\(29\) 8.87323 1.64772 0.823859 − 0.566795i \(-0.191817\pi\)
0.823859 + 0.566795i \(0.191817\pi\)
\(30\) 0 0
\(31\) 9.75286 1.75166 0.875832 − 0.482615i \(-0.160313\pi\)
0.875832 + 0.482615i \(0.160313\pi\)
\(32\) −1.00000 −0.176777
\(33\) −13.0174 −2.26605
\(34\) 3.31637 0.568752
\(35\) 0 0
\(36\) 4.45216 0.742027
\(37\) −1.00000 −0.164399
\(38\) 1.85109 0.300287
\(39\) 10.6941 1.71242
\(40\) 0 0
\(41\) −5.06977 −0.791764 −0.395882 − 0.918301i \(-0.629561\pi\)
−0.395882 + 0.918301i \(0.629561\pi\)
\(42\) −11.3108 −1.74530
\(43\) 9.99446 1.52414 0.762070 − 0.647494i \(-0.224183\pi\)
0.762070 + 0.647494i \(0.224183\pi\)
\(44\) −4.76853 −0.718883
\(45\) 0 0
\(46\) −1.54229 −0.227399
\(47\) −4.82700 −0.704090 −0.352045 − 0.935983i \(-0.614514\pi\)
−0.352045 + 0.935983i \(0.614514\pi\)
\(48\) 2.72987 0.394022
\(49\) 10.1675 1.45249
\(50\) 0 0
\(51\) −9.05324 −1.26771
\(52\) 3.91744 0.543251
\(53\) 5.13611 0.705499 0.352750 − 0.935718i \(-0.385247\pi\)
0.352750 + 0.935718i \(0.385247\pi\)
\(54\) −3.96421 −0.539461
\(55\) 0 0
\(56\) −4.14336 −0.553680
\(57\) −5.05324 −0.669317
\(58\) −8.87323 −1.16511
\(59\) −1.05324 −0.137120 −0.0685598 − 0.997647i \(-0.521840\pi\)
−0.0685598 + 0.997647i \(0.521840\pi\)
\(60\) 0 0
\(61\) −4.14422 −0.530613 −0.265306 − 0.964164i \(-0.585473\pi\)
−0.265306 + 0.964164i \(0.585473\pi\)
\(62\) −9.75286 −1.23861
\(63\) 18.4469 2.32410
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 13.0174 1.60234
\(67\) −1.00811 −0.123160 −0.0615800 − 0.998102i \(-0.519614\pi\)
−0.0615800 + 0.998102i \(0.519614\pi\)
\(68\) −3.31637 −0.402169
\(69\) 4.21025 0.506855
\(70\) 0 0
\(71\) 6.45248 0.765768 0.382884 − 0.923796i \(-0.374931\pi\)
0.382884 + 0.923796i \(0.374931\pi\)
\(72\) −4.45216 −0.524693
\(73\) −10.7714 −1.26070 −0.630349 − 0.776312i \(-0.717088\pi\)
−0.630349 + 0.776312i \(0.717088\pi\)
\(74\) 1.00000 0.116248
\(75\) 0 0
\(76\) −1.85109 −0.212335
\(77\) −19.7578 −2.25161
\(78\) −10.6941 −1.21087
\(79\) −1.19856 −0.134849 −0.0674243 − 0.997724i \(-0.521478\pi\)
−0.0674243 + 0.997724i \(0.521478\pi\)
\(80\) 0 0
\(81\) −2.53473 −0.281636
\(82\) 5.06977 0.559862
\(83\) −10.6932 −1.17373 −0.586867 − 0.809683i \(-0.699639\pi\)
−0.586867 + 0.809683i \(0.699639\pi\)
\(84\) 11.3108 1.23411
\(85\) 0 0
\(86\) −9.99446 −1.07773
\(87\) 24.2227 2.59695
\(88\) 4.76853 0.508327
\(89\) 7.29569 0.773342 0.386671 − 0.922218i \(-0.373625\pi\)
0.386671 + 0.922218i \(0.373625\pi\)
\(90\) 0 0
\(91\) 16.2314 1.70151
\(92\) 1.54229 0.160795
\(93\) 26.6240 2.76078
\(94\) 4.82700 0.497867
\(95\) 0 0
\(96\) −2.72987 −0.278616
\(97\) −14.8988 −1.51274 −0.756371 − 0.654143i \(-0.773030\pi\)
−0.756371 + 0.654143i \(0.773030\pi\)
\(98\) −10.1675 −1.02707
\(99\) −21.2303 −2.13372
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 1850.2.a.bd.1.5 5
5.2 odd 4 370.2.b.d.149.1 ✓ 10
5.3 odd 4 370.2.b.d.149.10 yes 10
5.4 even 2 1850.2.a.be.1.1 5
15.2 even 4 3330.2.d.p.1999.10 10
15.8 even 4 3330.2.d.p.1999.5 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
370.2.b.d.149.1 ✓ 10 5.2 odd 4
370.2.b.d.149.10 yes 10 5.3 odd 4
1850.2.a.bd.1.5 5 1.1 even 1 trivial
1850.2.a.be.1.1 5 5.4 even 2
3330.2.d.p.1999.5 10 15.8 even 4
3330.2.d.p.1999.10 10 15.2 even 4