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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1200,4,Mod(49,1200)] 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("1200.49"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1200, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 1])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 1200 = 2^{4} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1200.f (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,0,0,0,0,0,-18,0,-72,0,0,0,0,0,0,0,280] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(19)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(70.8022920069\)
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: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 60)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 49.2
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 1200.49
Dual form 1200.4.f.e.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)
 
\(f(q)\) \(=\) \(q+3.00000i q^{3} +32.0000i q^{7} -9.00000 q^{9} -36.0000 q^{11} -10.0000i q^{13} +78.0000i q^{17} +140.000 q^{19} -96.0000 q^{21} +192.000i q^{23} -27.0000i q^{27} -6.00000 q^{29} +16.0000 q^{31} -108.000i q^{33} +34.0000i q^{37} +30.0000 q^{39} -390.000 q^{41} +52.0000i q^{43} +408.000i q^{47} -681.000 q^{49} -234.000 q^{51} -114.000i q^{53} +420.000i q^{57} +516.000 q^{59} -58.0000 q^{61} -288.000i q^{63} -892.000i q^{67} -576.000 q^{69} +120.000 q^{71} -646.000i q^{73} -1152.00i q^{77} -1168.00 q^{79} +81.0000 q^{81} +732.000i q^{83} -18.0000i q^{87} +1590.00 q^{89} +320.000 q^{91} +48.0000i q^{93} -194.000i q^{97} +324.000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 18 q^{9} - 72 q^{11} + 280 q^{19} - 192 q^{21} - 12 q^{29} + 32 q^{31} + 60 q^{39} - 780 q^{41} - 1362 q^{49} - 468 q^{51} + 1032 q^{59} - 116 q^{61} - 1152 q^{69} + 240 q^{71} - 2336 q^{79} + 162 q^{81}+ \cdots + 648 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1200\mathbb{Z}\right)^\times\).

\(n\) \(401\) \(577\) \(751\) \(901\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 3.00000i 0.577350i
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 32.0000i 1.72784i 0.503631 + 0.863919i \(0.331997\pi\)
−0.503631 + 0.863919i \(0.668003\pi\)
\(8\) 0 0
\(9\) −9.00000 −0.333333
\(10\) 0 0
\(11\) −36.0000 −0.986764 −0.493382 − 0.869813i \(-0.664240\pi\)
−0.493382 + 0.869813i \(0.664240\pi\)
\(12\) 0 0
\(13\) − 10.0000i − 0.213346i −0.994294 − 0.106673i \(-0.965980\pi\)
0.994294 − 0.106673i \(-0.0340198\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 78.0000i 1.11281i 0.830911 + 0.556405i \(0.187820\pi\)
−0.830911 + 0.556405i \(0.812180\pi\)
\(18\) 0 0
\(19\) 140.000 1.69043 0.845216 − 0.534425i \(-0.179472\pi\)
0.845216 + 0.534425i \(0.179472\pi\)
\(20\) 0 0
\(21\) −96.0000 −0.997567
\(22\) 0 0
\(23\) 192.000i 1.74064i 0.492485 + 0.870321i \(0.336089\pi\)
−0.492485 + 0.870321i \(0.663911\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) − 27.0000i − 0.192450i
\(28\) 0 0
\(29\) −6.00000 −0.0384197 −0.0192099 − 0.999815i \(-0.506115\pi\)
−0.0192099 + 0.999815i \(0.506115\pi\)
\(30\) 0 0
\(31\) 16.0000 0.0926995 0.0463498 − 0.998925i \(-0.485241\pi\)
0.0463498 + 0.998925i \(0.485241\pi\)
\(32\) 0 0
\(33\) − 108.000i − 0.569709i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 34.0000i 0.151069i 0.997143 + 0.0755347i \(0.0240664\pi\)
−0.997143 + 0.0755347i \(0.975934\pi\)
\(38\) 0 0
\(39\) 30.0000 0.123176
\(40\) 0 0
\(41\) −390.000 −1.48556 −0.742778 − 0.669538i \(-0.766492\pi\)
−0.742778 + 0.669538i \(0.766492\pi\)
\(42\) 0 0
\(43\) 52.0000i 0.184417i 0.995740 + 0.0922084i \(0.0293926\pi\)
−0.995740 + 0.0922084i \(0.970607\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 408.000i 1.26623i 0.774057 + 0.633116i \(0.218224\pi\)
−0.774057 + 0.633116i \(0.781776\pi\)
\(48\) 0 0
\(49\) −681.000 −1.98542
\(50\) 0 0
\(51\) −234.000 −0.642481
\(52\) 0 0
\(53\) − 114.000i − 0.295455i −0.989028 − 0.147727i \(-0.952804\pi\)
0.989028 − 0.147727i \(-0.0471958\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 420.000i 0.975971i
\(58\) 0 0
\(59\) 516.000 1.13860 0.569301 − 0.822129i \(-0.307214\pi\)
0.569301 + 0.822129i \(0.307214\pi\)
\(60\) 0 0
\(61\) −58.0000 −0.121740 −0.0608700 − 0.998146i \(-0.519388\pi\)
−0.0608700 + 0.998146i \(0.519388\pi\)
\(62\) 0 0
\(63\) − 288.000i − 0.575946i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) − 892.000i − 1.62649i −0.581918 − 0.813247i \(-0.697698\pi\)
0.581918 − 0.813247i \(-0.302302\pi\)
\(68\) 0 0
\(69\) −576.000 −1.00496
\(70\) 0 0
\(71\) 120.000 0.200583 0.100291 − 0.994958i \(-0.468022\pi\)
0.100291 + 0.994958i \(0.468022\pi\)
\(72\) 0 0
\(73\) − 646.000i − 1.03573i −0.855461 − 0.517867i \(-0.826726\pi\)
0.855461 − 0.517867i \(-0.173274\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) − 1152.00i − 1.70497i
\(78\) 0 0
\(79\) −1168.00 −1.66342 −0.831711 − 0.555209i \(-0.812638\pi\)
−0.831711 + 0.555209i \(0.812638\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 0 0
\(83\) 732.000i 0.968041i 0.875057 + 0.484021i \(0.160824\pi\)
−0.875057 + 0.484021i \(0.839176\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) − 18.0000i − 0.0221816i
\(88\) 0 0
\(89\) 1590.00 1.89370 0.946852 − 0.321669i \(-0.104244\pi\)
0.946852 + 0.321669i \(0.104244\pi\)
\(90\) 0 0
\(91\) 320.000 0.368628
\(92\) 0 0
\(93\) 48.0000i 0.0535201i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) − 194.000i − 0.203069i −0.994832 − 0.101535i \(-0.967625\pi\)
0.994832 − 0.101535i \(-0.0323753\pi\)
\(98\) 0 0
\(99\) 324.000 0.328921
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 1200.4.f.e.49.2 2
4.3 odd 2 300.4.d.d.49.1 2
5.2 odd 4 240.4.a.j.1.1 1
5.3 odd 4 1200.4.a.s.1.1 1
5.4 even 2 inner 1200.4.f.e.49.1 2
12.11 even 2 900.4.d.b.649.1 2
15.2 even 4 720.4.a.c.1.1 1
20.3 even 4 300.4.a.e.1.1 1
20.7 even 4 60.4.a.b.1.1 ✓ 1
20.19 odd 2 300.4.d.d.49.2 2
40.27 even 4 960.4.a.bb.1.1 1
40.37 odd 4 960.4.a.a.1.1 1
60.23 odd 4 900.4.a.b.1.1 1
60.47 odd 4 180.4.a.c.1.1 1
60.59 even 2 900.4.d.b.649.2 2
180.7 even 12 1620.4.i.a.1081.1 2
180.47 odd 12 1620.4.i.g.1081.1 2
180.67 even 12 1620.4.i.a.541.1 2
180.167 odd 12 1620.4.i.g.541.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
60.4.a.b.1.1 ✓ 1 20.7 even 4
180.4.a.c.1.1 1 60.47 odd 4
240.4.a.j.1.1 1 5.2 odd 4
300.4.a.e.1.1 1 20.3 even 4
300.4.d.d.49.1 2 4.3 odd 2
300.4.d.d.49.2 2 20.19 odd 2
720.4.a.c.1.1 1 15.2 even 4
900.4.a.b.1.1 1 60.23 odd 4
900.4.d.b.649.1 2 12.11 even 2
900.4.d.b.649.2 2 60.59 even 2
960.4.a.a.1.1 1 40.37 odd 4
960.4.a.bb.1.1 1 40.27 even 4
1200.4.a.s.1.1 1 5.3 odd 4
1200.4.f.e.49.1 2 5.4 even 2 inner
1200.4.f.e.49.2 2 1.1 even 1 trivial
1620.4.i.a.541.1 2 180.67 even 12
1620.4.i.a.1081.1 2 180.7 even 12
1620.4.i.g.541.1 2 180.167 odd 12
1620.4.i.g.1081.1 2 180.47 odd 12