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 \(-2.62545\) 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)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +2.62545 q^{3} +1.00000 q^{4} +2.62545 q^{6} -1.83227 q^{7} +1.00000 q^{8} +3.89300 q^{9} +4.19017 q^{11} +2.62545 q^{12} -0.369454 q^{13} -1.83227 q^{14} +1.00000 q^{16} -5.08317 q^{17} +3.89300 q^{18} +3.55963 q^{19} -4.81053 q^{21} +4.19017 q^{22} +5.62036 q^{23} +2.62545 q^{24} -0.369454 q^{26} +2.34453 q^{27} -1.83227 q^{28} +1.20681 q^{29} +10.1030 q^{31} +1.00000 q^{32} +11.0011 q^{33} -5.08317 q^{34} +3.89300 q^{36} +1.00000 q^{37} +3.55963 q^{38} -0.969984 q^{39} -8.01447 q^{41} -4.81053 q^{42} -2.27264 q^{43} +4.19017 q^{44} +5.62036 q^{46} +10.9154 q^{47} +2.62545 q^{48} -3.64280 q^{49} -13.3456 q^{51} -0.369454 q^{52} -9.94355 q^{53} +2.34453 q^{54} -1.83227 q^{56} +9.34563 q^{57} +1.20681 q^{58} -5.34563 q^{59} -9.79428 q^{61} +10.1030 q^{62} -7.13302 q^{63} +1.00000 q^{64} +11.0011 q^{66} +1.85073 q^{67} -5.08317 q^{68} +14.7560 q^{69} +2.86038 q^{71} +3.89300 q^{72} -8.09942 q^{73} +1.00000 q^{74} +3.55963 q^{76} -7.67751 q^{77} -0.969984 q^{78} +6.06361 q^{79} -5.52355 q^{81} -8.01447 q^{82} -8.93200 q^{83} -4.81053 q^{84} -2.27264 q^{86} +3.16843 q^{87} +4.19017 q^{88} +11.4773 q^{89} +0.676938 q^{91} +5.62036 q^{92} +26.5249 q^{93} +10.9154 q^{94} +2.62545 q^{96} +6.05864 q^{97} -3.64280 q^{98} +16.3123 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q + 5 q^{2} + 5 q^{4} - q^{7} + 5 q^{8} + 3 q^{9} + 3 q^{11} - 6 q^{13} - q^{14} + 5 q^{16} + 9 q^{17} + 3 q^{18} + 4 q^{19} + 16 q^{21} + 3 q^{22} + 6 q^{23} - 6 q^{26} - q^{28} + 11 q^{29} + 23 q^{31}+ \cdots - 11 q^{99}+O(q^{100}) \) 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
\(3\) 2.62545 1.51581 0.757903 − 0.652367i \(-0.226224\pi\)
0.757903 + 0.652367i \(0.226224\pi\)
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) 2.62545 1.07184
\(7\) −1.83227 −0.692532 −0.346266 − 0.938136i \(-0.612551\pi\)
−0.346266 + 0.938136i \(0.612551\pi\)
\(8\) 1.00000 0.353553
\(9\) 3.89300 1.29767
\(10\) 0 0
\(11\) 4.19017 1.26338 0.631692 − 0.775219i \(-0.282361\pi\)
0.631692 + 0.775219i \(0.282361\pi\)
\(12\) 2.62545 0.757903
\(13\) −0.369454 −0.102468 −0.0512340 − 0.998687i \(-0.516315\pi\)
−0.0512340 + 0.998687i \(0.516315\pi\)
\(14\) −1.83227 −0.489694
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −5.08317 −1.23285 −0.616425 − 0.787414i \(-0.711420\pi\)
−0.616425 + 0.787414i \(0.711420\pi\)
\(18\) 3.89300 0.917589
\(19\) 3.55963 0.816634 0.408317 − 0.912840i \(-0.366116\pi\)
0.408317 + 0.912840i \(0.366116\pi\)
\(20\) 0 0
\(21\) −4.81053 −1.04974
\(22\) 4.19017 0.893348
\(23\) 5.62036 1.17193 0.585963 − 0.810338i \(-0.300716\pi\)
0.585963 + 0.810338i \(0.300716\pi\)
\(24\) 2.62545 0.535918
\(25\) 0 0
\(26\) −0.369454 −0.0724559
\(27\) 2.34453 0.451205
\(28\) −1.83227 −0.346266
\(29\) 1.20681 0.224100 0.112050 − 0.993703i \(-0.464258\pi\)
0.112050 + 0.993703i \(0.464258\pi\)
\(30\) 0 0
\(31\) 10.1030 1.81455 0.907276 − 0.420535i \(-0.138158\pi\)
0.907276 + 0.420535i \(0.138158\pi\)
\(32\) 1.00000 0.176777
\(33\) 11.0011 1.91504
\(34\) −5.08317 −0.871757
\(35\) 0 0
\(36\) 3.89300 0.648833
\(37\) 1.00000 0.164399
\(38\) 3.55963 0.577447
\(39\) −0.969984 −0.155322
\(40\) 0 0
\(41\) −8.01447 −1.25165 −0.625825 − 0.779964i \(-0.715238\pi\)
−0.625825 + 0.779964i \(0.715238\pi\)
\(42\) −4.81053 −0.742281
\(43\) −2.27264 −0.346575 −0.173287 − 0.984871i \(-0.555439\pi\)
−0.173287 + 0.984871i \(0.555439\pi\)
\(44\) 4.19017 0.631692
\(45\) 0 0
\(46\) 5.62036 0.828677
\(47\) 10.9154 1.59218 0.796090 − 0.605178i \(-0.206898\pi\)
0.796090 + 0.605178i \(0.206898\pi\)
\(48\) 2.62545 0.378951
\(49\) −3.64280 −0.520400
\(50\) 0 0
\(51\) −13.3456 −1.86876
\(52\) −0.369454 −0.0512340
\(53\) −9.94355 −1.36585 −0.682926 − 0.730488i \(-0.739293\pi\)
−0.682926 + 0.730488i \(0.739293\pi\)
\(54\) 2.34453 0.319050
\(55\) 0 0
\(56\) −1.83227 −0.244847
\(57\) 9.34563 1.23786
\(58\) 1.20681 0.158463
\(59\) −5.34563 −0.695941 −0.347971 − 0.937505i \(-0.613129\pi\)
−0.347971 + 0.937505i \(0.613129\pi\)
\(60\) 0 0
\(61\) −9.79428 −1.25403 −0.627015 − 0.779008i \(-0.715723\pi\)
−0.627015 + 0.779008i \(0.715723\pi\)
\(62\) 10.1030 1.28308
\(63\) −7.13302 −0.898676
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 11.0011 1.35414
\(67\) 1.85073 0.226103 0.113052 − 0.993589i \(-0.463937\pi\)
0.113052 + 0.993589i \(0.463937\pi\)
\(68\) −5.08317 −0.616425
\(69\) 14.7560 1.77641
\(70\) 0 0
\(71\) 2.86038 0.339464 0.169732 − 0.985490i \(-0.445710\pi\)
0.169732 + 0.985490i \(0.445710\pi\)
\(72\) 3.89300 0.458795
\(73\) −8.09942 −0.947966 −0.473983 − 0.880534i \(-0.657184\pi\)
−0.473983 + 0.880534i \(0.657184\pi\)
\(74\) 1.00000 0.116248
\(75\) 0 0
\(76\) 3.55963 0.408317
\(77\) −7.67751 −0.874934
\(78\) −0.969984 −0.109829
\(79\) 6.06361 0.682209 0.341105 − 0.940025i \(-0.389199\pi\)
0.341105 + 0.940025i \(0.389199\pi\)
\(80\) 0 0
\(81\) −5.52355 −0.613727
\(82\) −8.01447 −0.885050
\(83\) −8.93200 −0.980414 −0.490207 − 0.871606i \(-0.663079\pi\)
−0.490207 + 0.871606i \(0.663079\pi\)
\(84\) −4.81053 −0.524872
\(85\) 0 0
\(86\) −2.27264 −0.245065
\(87\) 3.16843 0.339692
\(88\) 4.19017 0.446674
\(89\) 11.4773 1.21659 0.608295 − 0.793711i \(-0.291854\pi\)
0.608295 + 0.793711i \(0.291854\pi\)
\(90\) 0 0
\(91\) 0.676938 0.0709624
\(92\) 5.62036 0.585963
\(93\) 26.5249 2.75051
\(94\) 10.9154 1.12584
\(95\) 0 0
\(96\) 2.62545 0.267959
\(97\) 6.05864 0.615162 0.307581 − 0.951522i \(-0.400480\pi\)
0.307581 + 0.951522i \(0.400480\pi\)
\(98\) −3.64280 −0.367978
\(99\) 16.3123 1.63945
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.be.1.5 5
5.2 odd 4 370.2.b.d.149.6 yes 10
5.3 odd 4 370.2.b.d.149.5 ✓ 10
5.4 even 2 1850.2.a.bd.1.1 5
15.2 even 4 3330.2.d.p.1999.3 10
15.8 even 4 3330.2.d.p.1999.8 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
370.2.b.d.149.5 ✓ 10 5.3 odd 4
370.2.b.d.149.6 yes 10 5.2 odd 4
1850.2.a.bd.1.1 5 5.4 even 2
1850.2.a.be.1.5 5 1.1 even 1 trivial
3330.2.d.p.1999.3 10 15.2 even 4
3330.2.d.p.1999.8 10 15.8 even 4