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Properties

Label 1850.2.b.o.149.1
Level $1850$
Weight $2$
Character 1850.149
Analytic conductor $14.772$
Analytic rank $0$
Dimension $6$
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] = [1850,2,Mod(149,1850)] 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("1850.149"); 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([1, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1850 = 2 \cdot 5^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1850.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,0,0,-6,0,0,0,0,-22,0,22] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(14.7723243739\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.3182656.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 2x^{3} + 25x^{2} - 10x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 370)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 149.1
Root \(-1.67298 + 1.67298i\) of defining polynomial
Character \(\chi\) \(=\) 1850.149
Dual form 1850.2.b.o.149.6

$q$-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)
 
\(f(q)\) \(=\) \(q-1.00000i q^{2} -3.34596i q^{3} -1.00000 q^{4} -3.34596 q^{6} +2.59774i q^{7} +1.00000i q^{8} -8.19547 q^{9} +4.74823 q^{11} +3.34596i q^{12} +6.69193i q^{13} +2.59774 q^{14} +1.00000 q^{16} +0.748228i q^{17} +8.19547i q^{18} +3.34596 q^{19} +8.69193 q^{21} -4.74823i q^{22} +1.49646i q^{23} +3.34596 q^{24} +6.69193 q^{26} +17.3839i q^{27} -2.59774i q^{28} -3.94370 q^{29} +7.79321 q^{31} -1.00000i q^{32} -15.8874i q^{33} +0.748228 q^{34} +8.19547 q^{36} +1.00000i q^{37} -3.34596i q^{38} +22.3909 q^{39} -6.44724 q^{41} -8.69193i q^{42} +1.94370i q^{43} -4.74823 q^{44} +1.49646 q^{46} -1.84951i q^{47} -3.34596i q^{48} +0.251772 q^{49} +2.50354 q^{51} -6.69193i q^{52} +10.4472i q^{53} +17.3839 q^{54} -2.59774 q^{56} -11.1955i q^{57} +3.94370i q^{58} -5.84951 q^{59} +7.94370 q^{61} -7.79321i q^{62} -21.2897i q^{63} -1.00000 q^{64} -15.8874 q^{66} -1.84951i q^{67} -0.748228i q^{68} +5.00709 q^{69} -3.88740 q^{71} -8.19547i q^{72} -7.49646i q^{73} +1.00000 q^{74} -3.34596 q^{76} +12.3346i q^{77} -22.3909i q^{78} +16.5414 q^{79} +33.5793 q^{81} +6.44724i q^{82} +15.2334i q^{83} -8.69193 q^{84} +1.94370 q^{86} +13.1955i q^{87} +4.74823i q^{88} -6.00000 q^{89} -17.3839 q^{91} -1.49646i q^{92} -26.0758i q^{93} -1.84951 q^{94} -3.34596 q^{96} +10.4472i q^{97} -0.251772i q^{98} -38.9140 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 6 q^{4} - 22 q^{9} + 22 q^{11} + 2 q^{14} + 6 q^{16} + 12 q^{21} + 10 q^{29} + 6 q^{31} - 2 q^{34} + 22 q^{36} + 80 q^{39} - 18 q^{41} - 22 q^{44} - 4 q^{46} + 8 q^{49} + 28 q^{51} + 24 q^{54} - 2 q^{56}+ \cdots - 38 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1850\mathbb{Z}\right)^\times\).

\(n\) \(1001\) \(1777\)
\(\chi(n)\) \(1\) \(-1\)

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
\(3\) − 3.34596i − 1.93179i −0.258929 − 0.965896i \(-0.583370\pi\)
0.258929 − 0.965896i \(-0.416630\pi\)
\(4\) −1.00000 −0.500000
\(5\) 0 0
\(6\) −3.34596 −1.36598
\(7\) 2.59774i 0.981852i 0.871201 + 0.490926i \(0.163341\pi\)
−0.871201 + 0.490926i \(0.836659\pi\)
\(8\) 1.00000i 0.353553i
\(9\) −8.19547 −2.73182
\(10\) 0 0
\(11\) 4.74823 1.43164 0.715822 − 0.698282i \(-0.246052\pi\)
0.715822 + 0.698282i \(0.246052\pi\)
\(12\) 3.34596i 0.965896i
\(13\) 6.69193i 1.85601i 0.372572 + 0.928003i \(0.378476\pi\)
−0.372572 + 0.928003i \(0.621524\pi\)
\(14\) 2.59774 0.694274
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 0.748228i 0.181472i 0.995875 + 0.0907360i \(0.0289219\pi\)
−0.995875 + 0.0907360i \(0.971078\pi\)
\(18\) 8.19547i 1.93169i
\(19\) 3.34596 0.767617 0.383808 − 0.923413i \(-0.374612\pi\)
0.383808 + 0.923413i \(0.374612\pi\)
\(20\) 0 0
\(21\) 8.69193 1.89673
\(22\) − 4.74823i − 1.01233i
\(23\) 1.49646i 0.312033i 0.987754 + 0.156016i \(0.0498653\pi\)
−0.987754 + 0.156016i \(0.950135\pi\)
\(24\) 3.34596 0.682992
\(25\) 0 0
\(26\) 6.69193 1.31239
\(27\) 17.3839i 3.34552i
\(28\) − 2.59774i − 0.490926i
\(29\) −3.94370 −0.732326 −0.366163 − 0.930551i \(-0.619329\pi\)
−0.366163 + 0.930551i \(0.619329\pi\)
\(30\) 0 0
\(31\) 7.79321 1.39970 0.699851 − 0.714289i \(-0.253250\pi\)
0.699851 + 0.714289i \(0.253250\pi\)
\(32\) − 1.00000i − 0.176777i
\(33\) − 15.8874i − 2.76564i
\(34\) 0.748228 0.128320
\(35\) 0 0
\(36\) 8.19547 1.36591
\(37\) 1.00000i 0.164399i
\(38\) − 3.34596i − 0.542787i
\(39\) 22.3909 3.58542
\(40\) 0 0
\(41\) −6.44724 −1.00689 −0.503445 − 0.864027i \(-0.667934\pi\)
−0.503445 + 0.864027i \(0.667934\pi\)
\(42\) − 8.69193i − 1.34119i
\(43\) 1.94370i 0.296411i 0.988957 + 0.148206i \(0.0473497\pi\)
−0.988957 + 0.148206i \(0.952650\pi\)
\(44\) −4.74823 −0.715822
\(45\) 0 0
\(46\) 1.49646 0.220640
\(47\) − 1.84951i − 0.269778i −0.990861 − 0.134889i \(-0.956932\pi\)
0.990861 − 0.134889i \(-0.0430678\pi\)
\(48\) − 3.34596i − 0.482948i
\(49\) 0.251772 0.0359674
\(50\) 0 0
\(51\) 2.50354 0.350566
\(52\) − 6.69193i − 0.928003i
\(53\) 10.4472i 1.43504i 0.696538 + 0.717520i \(0.254723\pi\)
−0.696538 + 0.717520i \(0.745277\pi\)
\(54\) 17.3839 2.36564
\(55\) 0 0
\(56\) −2.59774 −0.347137
\(57\) − 11.1955i − 1.48288i
\(58\) 3.94370i 0.517833i
\(59\) −5.84951 −0.761541 −0.380770 − 0.924670i \(-0.624341\pi\)
−0.380770 + 0.924670i \(0.624341\pi\)
\(60\) 0 0
\(61\) 7.94370 1.01709 0.508543 − 0.861036i \(-0.330184\pi\)
0.508543 + 0.861036i \(0.330184\pi\)
\(62\) − 7.79321i − 0.989738i
\(63\) − 21.2897i − 2.68225i
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) −15.8874 −1.95560
\(67\) − 1.84951i − 0.225953i −0.993598 − 0.112977i \(-0.963961\pi\)
0.993598 − 0.112977i \(-0.0360385\pi\)
\(68\) − 0.748228i − 0.0907360i
\(69\) 5.00709 0.602783
\(70\) 0 0
\(71\) −3.88740 −0.461349 −0.230675 − 0.973031i \(-0.574093\pi\)
−0.230675 + 0.973031i \(0.574093\pi\)
\(72\) − 8.19547i − 0.965845i
\(73\) − 7.49646i − 0.877394i −0.898635 − 0.438697i \(-0.855440\pi\)
0.898635 − 0.438697i \(-0.144560\pi\)
\(74\) 1.00000 0.116248
\(75\) 0 0
\(76\) −3.34596 −0.383808
\(77\) 12.3346i 1.40566i
\(78\) − 22.3909i − 2.53527i
\(79\) 16.5414 1.86106 0.930528 − 0.366220i \(-0.119348\pi\)
0.930528 + 0.366220i \(0.119348\pi\)
\(80\) 0 0
\(81\) 33.5793 3.73104
\(82\) 6.44724i 0.711979i
\(83\) 15.2334i 1.67208i 0.548670 + 0.836039i \(0.315134\pi\)
−0.548670 + 0.836039i \(0.684866\pi\)
\(84\) −8.69193 −0.948367
\(85\) 0 0
\(86\) 1.94370 0.209594
\(87\) 13.1955i 1.41470i
\(88\) 4.74823i 0.506163i
\(89\) −6.00000 −0.635999 −0.317999 − 0.948091i \(-0.603011\pi\)
−0.317999 + 0.948091i \(0.603011\pi\)
\(90\) 0 0
\(91\) −17.3839 −1.82232
\(92\) − 1.49646i − 0.156016i
\(93\) − 26.0758i − 2.70393i
\(94\) −1.84951 −0.190762
\(95\) 0 0
\(96\) −3.34596 −0.341496
\(97\) 10.4472i 1.06076i 0.847761 + 0.530378i \(0.177950\pi\)
−0.847761 + 0.530378i \(0.822050\pi\)
\(98\) − 0.251772i − 0.0254328i
\(99\) −38.9140 −3.91100
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.b.o.149.1 6
5.2 odd 4 370.2.a.g.1.1 ✓ 3
5.3 odd 4 1850.2.a.z.1.3 3
5.4 even 2 inner 1850.2.b.o.149.6 6
15.2 even 4 3330.2.a.bg.1.1 3
20.7 even 4 2960.2.a.u.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
370.2.a.g.1.1 ✓ 3 5.2 odd 4
1850.2.a.z.1.3 3 5.3 odd 4
1850.2.b.o.149.1 6 1.1 even 1 trivial
1850.2.b.o.149.6 6 5.4 even 2 inner
2960.2.a.u.1.3 3 20.7 even 4
3330.2.a.bg.1.1 3 15.2 even 4