Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [75,4,Mod(49,75)] 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("75.49"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(75, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 75 = 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 75.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,14] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(4)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(4.42514325043\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 15)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 49.2
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 75.49
Dual form 75.4.b.b.49.1

$q$-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)
 
\(f(q)\) \(=\) \(q+1.00000i q^{2} -3.00000i q^{3} +7.00000 q^{4} +3.00000 q^{6} -24.0000i q^{7} +15.0000i q^{8} -9.00000 q^{9} +52.0000 q^{11} -21.0000i q^{12} -22.0000i q^{13} +24.0000 q^{14} +41.0000 q^{16} -14.0000i q^{17} -9.00000i q^{18} +20.0000 q^{19} -72.0000 q^{21} +52.0000i q^{22} +168.000i q^{23} +45.0000 q^{24} +22.0000 q^{26} +27.0000i q^{27} -168.000i q^{28} -230.000 q^{29} -288.000 q^{31} +161.000i q^{32} -156.000i q^{33} +14.0000 q^{34} -63.0000 q^{36} -34.0000i q^{37} +20.0000i q^{38} -66.0000 q^{39} +122.000 q^{41} -72.0000i q^{42} +188.000i q^{43} +364.000 q^{44} -168.000 q^{46} +256.000i q^{47} -123.000i q^{48} -233.000 q^{49} -42.0000 q^{51} -154.000i q^{52} +338.000i q^{53} -27.0000 q^{54} +360.000 q^{56} -60.0000i q^{57} -230.000i q^{58} -100.000 q^{59} +742.000 q^{61} -288.000i q^{62} +216.000i q^{63} +167.000 q^{64} +156.000 q^{66} -84.0000i q^{67} -98.0000i q^{68} +504.000 q^{69} -328.000 q^{71} -135.000i q^{72} +38.0000i q^{73} +34.0000 q^{74} +140.000 q^{76} -1248.00i q^{77} -66.0000i q^{78} +240.000 q^{79} +81.0000 q^{81} +122.000i q^{82} -1212.00i q^{83} -504.000 q^{84} -188.000 q^{86} +690.000i q^{87} +780.000i q^{88} -330.000 q^{89} -528.000 q^{91} +1176.00i q^{92} +864.000i q^{93} -256.000 q^{94} +483.000 q^{96} +866.000i q^{97} -233.000i q^{98} -468.000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 14 q^{4} + 6 q^{6} - 18 q^{9} + 104 q^{11} + 48 q^{14} + 82 q^{16} + 40 q^{19} - 144 q^{21} + 90 q^{24} + 44 q^{26} - 460 q^{29} - 576 q^{31} + 28 q^{34} - 126 q^{36} - 132 q^{39} + 244 q^{41} + 728 q^{44}+ \cdots - 936 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/75\mathbb{Z}\right)^\times\).

\(n\) \(26\) \(52\)
\(\chi(n)\) \(1\) \(-1\)

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 0.353553i 0.984251 + 0.176777i \(0.0565670\pi\)
−0.984251 + 0.176777i \(0.943433\pi\)
\(3\) − 3.00000i − 0.577350i
\(4\) 7.00000 0.875000
\(5\) 0 0
\(6\) 3.00000 0.204124
\(7\) − 24.0000i − 1.29588i −0.761692 − 0.647939i \(-0.775631\pi\)
0.761692 − 0.647939i \(-0.224369\pi\)
\(8\) 15.0000i 0.662913i
\(9\) −9.00000 −0.333333
\(10\) 0 0
\(11\) 52.0000 1.42533 0.712663 − 0.701506i \(-0.247489\pi\)
0.712663 + 0.701506i \(0.247489\pi\)
\(12\) − 21.0000i − 0.505181i
\(13\) − 22.0000i − 0.469362i −0.972072 − 0.234681i \(-0.924595\pi\)
0.972072 − 0.234681i \(-0.0754045\pi\)
\(14\) 24.0000 0.458162
\(15\) 0 0
\(16\) 41.0000 0.640625
\(17\) − 14.0000i − 0.199735i −0.995001 − 0.0998676i \(-0.968158\pi\)
0.995001 − 0.0998676i \(-0.0318419\pi\)
\(18\) − 9.00000i − 0.117851i
\(19\) 20.0000 0.241490 0.120745 − 0.992684i \(-0.461472\pi\)
0.120745 + 0.992684i \(0.461472\pi\)
\(20\) 0 0
\(21\) −72.0000 −0.748176
\(22\) 52.0000i 0.503929i
\(23\) 168.000i 1.52306i 0.648129 + 0.761531i \(0.275552\pi\)
−0.648129 + 0.761531i \(0.724448\pi\)
\(24\) 45.0000 0.382733
\(25\) 0 0
\(26\) 22.0000 0.165944
\(27\) 27.0000i 0.192450i
\(28\) − 168.000i − 1.13389i
\(29\) −230.000 −1.47276 −0.736378 − 0.676570i \(-0.763465\pi\)
−0.736378 + 0.676570i \(0.763465\pi\)
\(30\) 0 0
\(31\) −288.000 −1.66859 −0.834296 − 0.551317i \(-0.814125\pi\)
−0.834296 + 0.551317i \(0.814125\pi\)
\(32\) 161.000i 0.889408i
\(33\) − 156.000i − 0.822913i
\(34\) 14.0000 0.0706171
\(35\) 0 0
\(36\) −63.0000 −0.291667
\(37\) − 34.0000i − 0.151069i −0.997143 − 0.0755347i \(-0.975934\pi\)
0.997143 − 0.0755347i \(-0.0240664\pi\)
\(38\) 20.0000i 0.0853797i
\(39\) −66.0000 −0.270986
\(40\) 0 0
\(41\) 122.000 0.464712 0.232356 − 0.972631i \(-0.425357\pi\)
0.232356 + 0.972631i \(0.425357\pi\)
\(42\) − 72.0000i − 0.264520i
\(43\) 188.000i 0.666738i 0.942796 + 0.333369i \(0.108185\pi\)
−0.942796 + 0.333369i \(0.891815\pi\)
\(44\) 364.000 1.24716
\(45\) 0 0
\(46\) −168.000 −0.538484
\(47\) 256.000i 0.794499i 0.917711 + 0.397249i \(0.130035\pi\)
−0.917711 + 0.397249i \(0.869965\pi\)
\(48\) − 123.000i − 0.369865i
\(49\) −233.000 −0.679300
\(50\) 0 0
\(51\) −42.0000 −0.115317
\(52\) − 154.000i − 0.410691i
\(53\) 338.000i 0.875998i 0.898976 + 0.437999i \(0.144313\pi\)
−0.898976 + 0.437999i \(0.855687\pi\)
\(54\) −27.0000 −0.0680414
\(55\) 0 0
\(56\) 360.000 0.859054
\(57\) − 60.0000i − 0.139424i
\(58\) − 230.000i − 0.520698i
\(59\) −100.000 −0.220659 −0.110330 − 0.993895i \(-0.535191\pi\)
−0.110330 + 0.993895i \(0.535191\pi\)
\(60\) 0 0
\(61\) 742.000 1.55743 0.778716 − 0.627376i \(-0.215871\pi\)
0.778716 + 0.627376i \(0.215871\pi\)
\(62\) − 288.000i − 0.589936i
\(63\) 216.000i 0.431959i
\(64\) 167.000 0.326172
\(65\) 0 0
\(66\) 156.000 0.290944
\(67\) − 84.0000i − 0.153168i −0.997063 − 0.0765838i \(-0.975599\pi\)
0.997063 − 0.0765838i \(-0.0244013\pi\)
\(68\) − 98.0000i − 0.174768i
\(69\) 504.000 0.879340
\(70\) 0 0
\(71\) −328.000 −0.548260 −0.274130 − 0.961693i \(-0.588390\pi\)
−0.274130 + 0.961693i \(0.588390\pi\)
\(72\) − 135.000i − 0.220971i
\(73\) 38.0000i 0.0609255i 0.999536 + 0.0304628i \(0.00969810\pi\)
−0.999536 + 0.0304628i \(0.990302\pi\)
\(74\) 34.0000 0.0534111
\(75\) 0 0
\(76\) 140.000 0.211304
\(77\) − 1248.00i − 1.84705i
\(78\) − 66.0000i − 0.0958081i
\(79\) 240.000 0.341799 0.170899 − 0.985288i \(-0.445333\pi\)
0.170899 + 0.985288i \(0.445333\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 122.000i 0.164301i
\(83\) − 1212.00i − 1.60282i −0.598114 − 0.801411i \(-0.704083\pi\)
0.598114 − 0.801411i \(-0.295917\pi\)
\(84\) −504.000 −0.654654
\(85\) 0 0
\(86\) −188.000 −0.235727
\(87\) 690.000i 0.850296i
\(88\) 780.000i 0.944867i
\(89\) −330.000 −0.393033 −0.196516 − 0.980501i \(-0.562963\pi\)
−0.196516 + 0.980501i \(0.562963\pi\)
\(90\) 0 0
\(91\) −528.000 −0.608236
\(92\) 1176.00i 1.33268i
\(93\) 864.000i 0.963362i
\(94\) −256.000 −0.280898
\(95\) 0 0
\(96\) 483.000 0.513500
\(97\) 866.000i 0.906484i 0.891387 + 0.453242i \(0.149733\pi\)
−0.891387 + 0.453242i \(0.850267\pi\)
\(98\) − 233.000i − 0.240169i
\(99\) −468.000 −0.475109
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 75.4.b.b.49.2 2
3.2 odd 2 225.4.b.e.199.1 2
4.3 odd 2 1200.4.f.b.49.2 2
5.2 odd 4 75.4.a.b.1.1 1
5.3 odd 4 15.4.a.a.1.1 ✓ 1
5.4 even 2 inner 75.4.b.b.49.1 2
15.2 even 4 225.4.a.f.1.1 1
15.8 even 4 45.4.a.c.1.1 1
15.14 odd 2 225.4.b.e.199.2 2
20.3 even 4 240.4.a.e.1.1 1
20.7 even 4 1200.4.a.t.1.1 1
20.19 odd 2 1200.4.f.b.49.1 2
35.13 even 4 735.4.a.e.1.1 1
40.3 even 4 960.4.a.ba.1.1 1
40.13 odd 4 960.4.a.b.1.1 1
45.13 odd 12 405.4.e.g.136.1 2
45.23 even 12 405.4.e.i.136.1 2
45.38 even 12 405.4.e.i.271.1 2
45.43 odd 12 405.4.e.g.271.1 2
55.43 even 4 1815.4.a.e.1.1 1
60.23 odd 4 720.4.a.n.1.1 1
105.83 odd 4 2205.4.a.l.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
15.4.a.a.1.1 ✓ 1 5.3 odd 4
45.4.a.c.1.1 1 15.8 even 4
75.4.a.b.1.1 1 5.2 odd 4
75.4.b.b.49.1 2 5.4 even 2 inner
75.4.b.b.49.2 2 1.1 even 1 trivial
225.4.a.f.1.1 1 15.2 even 4
225.4.b.e.199.1 2 3.2 odd 2
225.4.b.e.199.2 2 15.14 odd 2
240.4.a.e.1.1 1 20.3 even 4
405.4.e.g.136.1 2 45.13 odd 12
405.4.e.g.271.1 2 45.43 odd 12
405.4.e.i.136.1 2 45.23 even 12
405.4.e.i.271.1 2 45.38 even 12
720.4.a.n.1.1 1 60.23 odd 4
735.4.a.e.1.1 1 35.13 even 4
960.4.a.b.1.1 1 40.13 odd 4
960.4.a.ba.1.1 1 40.3 even 4
1200.4.a.t.1.1 1 20.7 even 4
1200.4.f.b.49.1 2 20.19 odd 2
1200.4.f.b.49.2 2 4.3 odd 2
1815.4.a.e.1.1 1 55.43 even 4
2205.4.a.l.1.1 1 105.83 odd 4