Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [175,2,Mod(1,175)] 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("175.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(175, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 175 = 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 175.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,1,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(1.39738203537\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{10})^+\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-0.618034\) of defining polynomial
Character \(\chi\) \(=\) 175.1

$q$-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)
 
\(f(q)\) \(=\) \(q-0.618034 q^{2} -3.23607 q^{3} -1.61803 q^{4} +2.00000 q^{6} +1.00000 q^{7} +2.23607 q^{8} +7.47214 q^{9} -0.236068 q^{11} +5.23607 q^{12} +1.23607 q^{13} -0.618034 q^{14} +1.85410 q^{16} +2.47214 q^{17} -4.61803 q^{18} -4.47214 q^{19} -3.23607 q^{21} +0.145898 q^{22} +6.23607 q^{23} -7.23607 q^{24} -0.763932 q^{26} -14.4721 q^{27} -1.61803 q^{28} +5.00000 q^{29} +3.70820 q^{31} -5.61803 q^{32} +0.763932 q^{33} -1.52786 q^{34} -12.0902 q^{36} +3.00000 q^{37} +2.76393 q^{38} -4.00000 q^{39} +4.76393 q^{41} +2.00000 q^{42} +1.76393 q^{43} +0.381966 q^{44} -3.85410 q^{46} -2.00000 q^{47} -6.00000 q^{48} +1.00000 q^{49} -8.00000 q^{51} -2.00000 q^{52} +8.47214 q^{53} +8.94427 q^{54} +2.23607 q^{56} +14.4721 q^{57} -3.09017 q^{58} +11.7082 q^{59} -9.70820 q^{61} -2.29180 q^{62} +7.47214 q^{63} -0.236068 q^{64} -0.472136 q^{66} -4.23607 q^{67} -4.00000 q^{68} -20.1803 q^{69} +8.70820 q^{71} +16.7082 q^{72} -8.76393 q^{73} -1.85410 q^{74} +7.23607 q^{76} -0.236068 q^{77} +2.47214 q^{78} -11.1803 q^{79} +24.4164 q^{81} -2.94427 q^{82} -7.70820 q^{83} +5.23607 q^{84} -1.09017 q^{86} -16.1803 q^{87} -0.527864 q^{88} +17.2361 q^{89} +1.23607 q^{91} -10.0902 q^{92} -12.0000 q^{93} +1.23607 q^{94} +18.1803 q^{96} +5.23607 q^{97} -0.618034 q^{98} -1.76393 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{2} - 2 q^{3} - q^{4} + 4 q^{6} + 2 q^{7} + 6 q^{9} + 4 q^{11} + 6 q^{12} - 2 q^{13} + q^{14} - 3 q^{16} - 4 q^{17} - 7 q^{18} - 2 q^{21} + 7 q^{22} + 8 q^{23} - 10 q^{24} - 6 q^{26} - 20 q^{27}+ \cdots - 8 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.618034 −0.437016 −0.218508 − 0.975835i \(-0.570119\pi\)
−0.218508 + 0.975835i \(0.570119\pi\)
\(3\) −3.23607 −1.86834 −0.934172 − 0.356822i \(-0.883860\pi\)
−0.934172 + 0.356822i \(0.883860\pi\)
\(4\) −1.61803 −0.809017
\(5\) 0 0
\(6\) 2.00000 0.816497
\(7\) 1.00000 0.377964
\(8\) 2.23607 0.790569
\(9\) 7.47214 2.49071
\(10\) 0 0
\(11\) −0.236068 −0.0711772 −0.0355886 − 0.999367i \(-0.511331\pi\)
−0.0355886 + 0.999367i \(0.511331\pi\)
\(12\) 5.23607 1.51152
\(13\) 1.23607 0.342824 0.171412 − 0.985199i \(-0.445167\pi\)
0.171412 + 0.985199i \(0.445167\pi\)
\(14\) −0.618034 −0.165177
\(15\) 0 0
\(16\) 1.85410 0.463525
\(17\) 2.47214 0.599581 0.299791 − 0.954005i \(-0.403083\pi\)
0.299791 + 0.954005i \(0.403083\pi\)
\(18\) −4.61803 −1.08848
\(19\) −4.47214 −1.02598 −0.512989 − 0.858395i \(-0.671462\pi\)
−0.512989 + 0.858395i \(0.671462\pi\)
\(20\) 0 0
\(21\) −3.23607 −0.706168
\(22\) 0.145898 0.0311056
\(23\) 6.23607 1.30031 0.650155 − 0.759802i \(-0.274704\pi\)
0.650155 + 0.759802i \(0.274704\pi\)
\(24\) −7.23607 −1.47706
\(25\) 0 0
\(26\) −0.763932 −0.149819
\(27\) −14.4721 −2.78516
\(28\) −1.61803 −0.305780
\(29\) 5.00000 0.928477 0.464238 − 0.885710i \(-0.346328\pi\)
0.464238 + 0.885710i \(0.346328\pi\)
\(30\) 0 0
\(31\) 3.70820 0.666013 0.333007 − 0.942925i \(-0.391937\pi\)
0.333007 + 0.942925i \(0.391937\pi\)
\(32\) −5.61803 −0.993137
\(33\) 0.763932 0.132983
\(34\) −1.52786 −0.262027
\(35\) 0 0
\(36\) −12.0902 −2.01503
\(37\) 3.00000 0.493197 0.246598 − 0.969118i \(-0.420687\pi\)
0.246598 + 0.969118i \(0.420687\pi\)
\(38\) 2.76393 0.448369
\(39\) −4.00000 −0.640513
\(40\) 0 0
\(41\) 4.76393 0.744001 0.372001 − 0.928232i \(-0.378672\pi\)
0.372001 + 0.928232i \(0.378672\pi\)
\(42\) 2.00000 0.308607
\(43\) 1.76393 0.268997 0.134499 − 0.990914i \(-0.457058\pi\)
0.134499 + 0.990914i \(0.457058\pi\)
\(44\) 0.381966 0.0575835
\(45\) 0 0
\(46\) −3.85410 −0.568256
\(47\) −2.00000 −0.291730 −0.145865 − 0.989305i \(-0.546597\pi\)
−0.145865 + 0.989305i \(0.546597\pi\)
\(48\) −6.00000 −0.866025
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) −8.00000 −1.12022
\(52\) −2.00000 −0.277350
\(53\) 8.47214 1.16374 0.581869 − 0.813283i \(-0.302322\pi\)
0.581869 + 0.813283i \(0.302322\pi\)
\(54\) 8.94427 1.21716
\(55\) 0 0
\(56\) 2.23607 0.298807
\(57\) 14.4721 1.91688
\(58\) −3.09017 −0.405759
\(59\) 11.7082 1.52428 0.762139 − 0.647413i \(-0.224149\pi\)
0.762139 + 0.647413i \(0.224149\pi\)
\(60\) 0 0
\(61\) −9.70820 −1.24301 −0.621504 − 0.783411i \(-0.713478\pi\)
−0.621504 + 0.783411i \(0.713478\pi\)
\(62\) −2.29180 −0.291058
\(63\) 7.47214 0.941401
\(64\) −0.236068 −0.0295085
\(65\) 0 0
\(66\) −0.472136 −0.0581159
\(67\) −4.23607 −0.517518 −0.258759 − 0.965942i \(-0.583314\pi\)
−0.258759 + 0.965942i \(0.583314\pi\)
\(68\) −4.00000 −0.485071
\(69\) −20.1803 −2.42943
\(70\) 0 0
\(71\) 8.70820 1.03347 0.516737 − 0.856144i \(-0.327147\pi\)
0.516737 + 0.856144i \(0.327147\pi\)
\(72\) 16.7082 1.96908
\(73\) −8.76393 −1.02574 −0.512870 − 0.858466i \(-0.671418\pi\)
−0.512870 + 0.858466i \(0.671418\pi\)
\(74\) −1.85410 −0.215535
\(75\) 0 0
\(76\) 7.23607 0.830034
\(77\) −0.236068 −0.0269024
\(78\) 2.47214 0.279914
\(79\) −11.1803 −1.25789 −0.628943 − 0.777451i \(-0.716512\pi\)
−0.628943 + 0.777451i \(0.716512\pi\)
\(80\) 0 0
\(81\) 24.4164 2.71293
\(82\) −2.94427 −0.325140
\(83\) −7.70820 −0.846085 −0.423043 − 0.906110i \(-0.639038\pi\)
−0.423043 + 0.906110i \(0.639038\pi\)
\(84\) 5.23607 0.571302
\(85\) 0 0
\(86\) −1.09017 −0.117556
\(87\) −16.1803 −1.73471
\(88\) −0.527864 −0.0562705
\(89\) 17.2361 1.82702 0.913510 − 0.406817i \(-0.133361\pi\)
0.913510 + 0.406817i \(0.133361\pi\)
\(90\) 0 0
\(91\) 1.23607 0.129575
\(92\) −10.0902 −1.05197
\(93\) −12.0000 −1.24434
\(94\) 1.23607 0.127491
\(95\) 0 0
\(96\) 18.1803 1.85552
\(97\) 5.23607 0.531642 0.265821 − 0.964022i \(-0.414357\pi\)
0.265821 + 0.964022i \(0.414357\pi\)
\(98\) −0.618034 −0.0624309
\(99\) −1.76393 −0.177282
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 175.2.a.e.1.1 yes 2
3.2 odd 2 1575.2.a.n.1.2 2
4.3 odd 2 2800.2.a.bp.1.2 2
5.2 odd 4 175.2.b.c.99.2 4
5.3 odd 4 175.2.b.c.99.3 4
5.4 even 2 175.2.a.d.1.2 ✓ 2
7.6 odd 2 1225.2.a.u.1.1 2
15.2 even 4 1575.2.d.k.1324.3 4
15.8 even 4 1575.2.d.k.1324.2 4
15.14 odd 2 1575.2.a.s.1.1 2
20.3 even 4 2800.2.g.s.449.4 4
20.7 even 4 2800.2.g.s.449.1 4
20.19 odd 2 2800.2.a.bh.1.1 2
35.13 even 4 1225.2.b.k.99.3 4
35.27 even 4 1225.2.b.k.99.2 4
35.34 odd 2 1225.2.a.n.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
175.2.a.d.1.2 ✓ 2 5.4 even 2
175.2.a.e.1.1 yes 2 1.1 even 1 trivial
175.2.b.c.99.2 4 5.2 odd 4
175.2.b.c.99.3 4 5.3 odd 4
1225.2.a.n.1.2 2 35.34 odd 2
1225.2.a.u.1.1 2 7.6 odd 2
1225.2.b.k.99.2 4 35.27 even 4
1225.2.b.k.99.3 4 35.13 even 4
1575.2.a.n.1.2 2 3.2 odd 2
1575.2.a.s.1.1 2 15.14 odd 2
1575.2.d.k.1324.2 4 15.8 even 4
1575.2.d.k.1324.3 4 15.2 even 4
2800.2.a.bh.1.1 2 20.19 odd 2
2800.2.a.bp.1.2 2 4.3 odd 2
2800.2.g.s.449.1 4 20.7 even 4
2800.2.g.s.449.4 4 20.3 even 4