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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [2475,2,Mod(199,2475)] 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("2475.199"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2475, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 2475 = 3^{2} \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2475.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,2,0,0,0,0,0,0,-2,0,0,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(14)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(19.7629745003\)
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 99)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 199.1
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 2475.199
Dual form 2475.2.c.b.199.2

$q$-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)
 
\(f(q)\) \(=\) \(q-1.00000i q^{2} +1.00000 q^{4} -2.00000i q^{7} -3.00000i q^{8} -1.00000 q^{11} +2.00000i q^{13} -2.00000 q^{14} -1.00000 q^{16} +2.00000i q^{17} +6.00000 q^{19} +1.00000i q^{22} -4.00000i q^{23} +2.00000 q^{26} -2.00000i q^{28} +6.00000 q^{29} +4.00000 q^{31} -5.00000i q^{32} +2.00000 q^{34} -6.00000i q^{37} -6.00000i q^{38} -10.0000 q^{41} -6.00000i q^{43} -1.00000 q^{44} -4.00000 q^{46} -8.00000i q^{47} +3.00000 q^{49} +2.00000i q^{52} -6.00000 q^{56} -6.00000i q^{58} -4.00000 q^{59} -6.00000 q^{61} -4.00000i q^{62} -7.00000 q^{64} +8.00000i q^{67} +2.00000i q^{68} +2.00000i q^{73} -6.00000 q^{74} +6.00000 q^{76} +2.00000i q^{77} +10.0000 q^{79} +10.0000i q^{82} -12.0000i q^{83} -6.00000 q^{86} +3.00000i q^{88} +4.00000 q^{91} -4.00000i q^{92} -8.00000 q^{94} +2.00000i q^{97} -3.00000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{4} - 2 q^{11} - 4 q^{14} - 2 q^{16} + 12 q^{19} + 4 q^{26} + 12 q^{29} + 8 q^{31} + 4 q^{34} - 20 q^{41} - 2 q^{44} - 8 q^{46} + 6 q^{49} - 12 q^{56} - 8 q^{59} - 12 q^{61} - 14 q^{64} - 12 q^{74}+ \cdots - 16 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2475\mathbb{Z}\right)^\times\).

\(n\) \(551\) \(2026\) \(2377\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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 −0.935414 − 0.353553i \(-0.884973\pi\)
0.935414 − 0.353553i \(-0.115027\pi\)
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) 0 0
\(7\) − 2.00000i − 0.755929i −0.925820 − 0.377964i \(-0.876624\pi\)
0.925820 − 0.377964i \(-0.123376\pi\)
\(8\) − 3.00000i − 1.06066i
\(9\) 0 0
\(10\) 0 0
\(11\) −1.00000 −0.301511
\(12\) 0 0
\(13\) 2.00000i 0.554700i 0.960769 + 0.277350i \(0.0894562\pi\)
−0.960769 + 0.277350i \(0.910544\pi\)
\(14\) −2.00000 −0.534522
\(15\) 0 0
\(16\) −1.00000 −0.250000
\(17\) 2.00000i 0.485071i 0.970143 + 0.242536i \(0.0779791\pi\)
−0.970143 + 0.242536i \(0.922021\pi\)
\(18\) 0 0
\(19\) 6.00000 1.37649 0.688247 − 0.725476i \(-0.258380\pi\)
0.688247 + 0.725476i \(0.258380\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 1.00000i 0.213201i
\(23\) − 4.00000i − 0.834058i −0.908893 − 0.417029i \(-0.863071\pi\)
0.908893 − 0.417029i \(-0.136929\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 2.00000 0.392232
\(27\) 0 0
\(28\) − 2.00000i − 0.377964i
\(29\) 6.00000 1.11417 0.557086 − 0.830455i \(-0.311919\pi\)
0.557086 + 0.830455i \(0.311919\pi\)
\(30\) 0 0
\(31\) 4.00000 0.718421 0.359211 − 0.933257i \(-0.383046\pi\)
0.359211 + 0.933257i \(0.383046\pi\)
\(32\) − 5.00000i − 0.883883i
\(33\) 0 0
\(34\) 2.00000 0.342997
\(35\) 0 0
\(36\) 0 0
\(37\) − 6.00000i − 0.986394i −0.869918 − 0.493197i \(-0.835828\pi\)
0.869918 − 0.493197i \(-0.164172\pi\)
\(38\) − 6.00000i − 0.973329i
\(39\) 0 0
\(40\) 0 0
\(41\) −10.0000 −1.56174 −0.780869 − 0.624695i \(-0.785223\pi\)
−0.780869 + 0.624695i \(0.785223\pi\)
\(42\) 0 0
\(43\) − 6.00000i − 0.914991i −0.889212 − 0.457496i \(-0.848747\pi\)
0.889212 − 0.457496i \(-0.151253\pi\)
\(44\) −1.00000 −0.150756
\(45\) 0 0
\(46\) −4.00000 −0.589768
\(47\) − 8.00000i − 1.16692i −0.812142 − 0.583460i \(-0.801699\pi\)
0.812142 − 0.583460i \(-0.198301\pi\)
\(48\) 0 0
\(49\) 3.00000 0.428571
\(50\) 0 0
\(51\) 0 0
\(52\) 2.00000i 0.277350i
\(53\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −6.00000 −0.801784
\(57\) 0 0
\(58\) − 6.00000i − 0.787839i
\(59\) −4.00000 −0.520756 −0.260378 − 0.965507i \(-0.583847\pi\)
−0.260378 + 0.965507i \(0.583847\pi\)
\(60\) 0 0
\(61\) −6.00000 −0.768221 −0.384111 − 0.923287i \(-0.625492\pi\)
−0.384111 + 0.923287i \(0.625492\pi\)
\(62\) − 4.00000i − 0.508001i
\(63\) 0 0
\(64\) −7.00000 −0.875000
\(65\) 0 0
\(66\) 0 0
\(67\) 8.00000i 0.977356i 0.872464 + 0.488678i \(0.162521\pi\)
−0.872464 + 0.488678i \(0.837479\pi\)
\(68\) 2.00000i 0.242536i
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) 2.00000i 0.234082i 0.993127 + 0.117041i \(0.0373409\pi\)
−0.993127 + 0.117041i \(0.962659\pi\)
\(74\) −6.00000 −0.697486
\(75\) 0 0
\(76\) 6.00000 0.688247
\(77\) 2.00000i 0.227921i
\(78\) 0 0
\(79\) 10.0000 1.12509 0.562544 − 0.826767i \(-0.309823\pi\)
0.562544 + 0.826767i \(0.309823\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 10.0000i 1.10432i
\(83\) − 12.0000i − 1.31717i −0.752506 − 0.658586i \(-0.771155\pi\)
0.752506 − 0.658586i \(-0.228845\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −6.00000 −0.646997
\(87\) 0 0
\(88\) 3.00000i 0.319801i
\(89\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) 0 0
\(91\) 4.00000 0.419314
\(92\) − 4.00000i − 0.417029i
\(93\) 0 0
\(94\) −8.00000 −0.825137
\(95\) 0 0
\(96\) 0 0
\(97\) 2.00000i 0.203069i 0.994832 + 0.101535i \(0.0323753\pi\)
−0.994832 + 0.101535i \(0.967625\pi\)
\(98\) − 3.00000i − 0.303046i
\(99\) 0 0
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 2475.2.c.b.199.1 2
3.2 odd 2 2475.2.c.g.199.2 2
5.2 odd 4 2475.2.a.j.1.1 1
5.3 odd 4 99.2.a.a.1.1 ✓ 1
5.4 even 2 inner 2475.2.c.b.199.2 2
15.2 even 4 2475.2.a.c.1.1 1
15.8 even 4 99.2.a.c.1.1 yes 1
15.14 odd 2 2475.2.c.g.199.1 2
20.3 even 4 1584.2.a.b.1.1 1
35.13 even 4 4851.2.a.g.1.1 1
40.3 even 4 6336.2.a.cm.1.1 1
40.13 odd 4 6336.2.a.cl.1.1 1
45.13 odd 12 891.2.e.j.298.1 2
45.23 even 12 891.2.e.c.298.1 2
45.38 even 12 891.2.e.c.595.1 2
45.43 odd 12 891.2.e.j.595.1 2
55.43 even 4 1089.2.a.h.1.1 1
60.23 odd 4 1584.2.a.r.1.1 1
105.83 odd 4 4851.2.a.o.1.1 1
120.53 even 4 6336.2.a.b.1.1 1
120.83 odd 4 6336.2.a.f.1.1 1
165.98 odd 4 1089.2.a.d.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
99.2.a.a.1.1 ✓ 1 5.3 odd 4
99.2.a.c.1.1 yes 1 15.8 even 4
891.2.e.c.298.1 2 45.23 even 12
891.2.e.c.595.1 2 45.38 even 12
891.2.e.j.298.1 2 45.13 odd 12
891.2.e.j.595.1 2 45.43 odd 12
1089.2.a.d.1.1 1 165.98 odd 4
1089.2.a.h.1.1 1 55.43 even 4
1584.2.a.b.1.1 1 20.3 even 4
1584.2.a.r.1.1 1 60.23 odd 4
2475.2.a.c.1.1 1 15.2 even 4
2475.2.a.j.1.1 1 5.2 odd 4
2475.2.c.b.199.1 2 1.1 even 1 trivial
2475.2.c.b.199.2 2 5.4 even 2 inner
2475.2.c.g.199.1 2 15.14 odd 2
2475.2.c.g.199.2 2 3.2 odd 2
4851.2.a.g.1.1 1 35.13 even 4
4851.2.a.o.1.1 1 105.83 odd 4
6336.2.a.b.1.1 1 120.53 even 4
6336.2.a.f.1.1 1 120.83 odd 4
6336.2.a.cl.1.1 1 40.13 odd 4
6336.2.a.cm.1.1 1 40.3 even 4