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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [3552,2,Mod(2737,3552)] 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("3552.2737"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3552, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 3552 = 2^{5} \cdot 3 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3552.o (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(28.3628627980\)
Analytic rank: \(0\)
Dimension: \(76\)
Twist minimal: no (minimal twist has level 888)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 2737.7
Character \(\chi\) \(=\) 3552.2737
Dual form 3552.2.o.a.2737.8

$q$-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)
 
\(f(q)\) \(=\) \(q-1.00000i q^{3} +1.36027 q^{5} -1.05670 q^{7} -1.00000 q^{9} -6.25278i q^{11} +2.37440 q^{13} -1.36027i q^{15} +1.36964i q^{17} -1.72927 q^{19} +1.05670i q^{21} -0.0334155i q^{23} -3.14967 q^{25} +1.00000i q^{27} +4.15557 q^{29} -5.58043i q^{31} -6.25278 q^{33} -1.43739 q^{35} +(3.49132 - 4.98103i) q^{37} -2.37440i q^{39} +5.36694 q^{41} -3.49998 q^{43} -1.36027 q^{45} +2.11515 q^{47} -5.88338 q^{49} +1.36964 q^{51} +0.643686i q^{53} -8.50546i q^{55} +1.72927i q^{57} -10.1925 q^{59} -2.79008 q^{61} +1.05670 q^{63} +3.22981 q^{65} +2.63276i q^{67} -0.0334155 q^{69} -8.36661 q^{71} -4.99853 q^{73} +3.14967i q^{75} +6.60732i q^{77} +1.86283i q^{79} +1.00000 q^{81} -0.00717686i q^{83} +1.86307i q^{85} -4.15557i q^{87} -0.818212i q^{89} -2.50903 q^{91} -5.58043 q^{93} -2.35228 q^{95} -11.1763i q^{97} +6.25278i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 76 q - 8 q^{7} - 76 q^{9} + 84 q^{25} - 8 q^{41} + 60 q^{49} + 8 q^{63} + 48 q^{65} + 16 q^{71} + 24 q^{73} + 76 q^{81}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3552\mathbb{Z}\right)^\times\).

\(n\) \(223\) \(2369\) \(3073\) \(3109\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(-1\)

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
\(3\) − 1.00000i − 0.577350i
\(4\) 0 0
\(5\) 1.36027 0.608330 0.304165 − 0.952619i \(-0.401623\pi\)
0.304165 + 0.952619i \(0.401623\pi\)
\(6\) 0 0
\(7\) −1.05670 −0.399395 −0.199698 − 0.979858i \(-0.563996\pi\)
−0.199698 + 0.979858i \(0.563996\pi\)
\(8\) 0 0
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) − 6.25278i − 1.88529i −0.333803 − 0.942643i \(-0.608332\pi\)
0.333803 − 0.942643i \(-0.391668\pi\)
\(12\) 0 0
\(13\) 2.37440 0.658539 0.329270 − 0.944236i \(-0.393197\pi\)
0.329270 + 0.944236i \(0.393197\pi\)
\(14\) 0 0
\(15\) − 1.36027i − 0.351219i
\(16\) 0 0
\(17\) 1.36964i 0.332186i 0.986110 + 0.166093i \(0.0531151\pi\)
−0.986110 + 0.166093i \(0.946885\pi\)
\(18\) 0 0
\(19\) −1.72927 −0.396723 −0.198361 − 0.980129i \(-0.563562\pi\)
−0.198361 + 0.980129i \(0.563562\pi\)
\(20\) 0 0
\(21\) 1.05670i 0.230591i
\(22\) 0 0
\(23\) − 0.0334155i − 0.00696760i −0.999994 − 0.00348380i \(-0.998891\pi\)
0.999994 − 0.00348380i \(-0.00110893\pi\)
\(24\) 0 0
\(25\) −3.14967 −0.629935
\(26\) 0 0
\(27\) 1.00000i 0.192450i
\(28\) 0 0
\(29\) 4.15557 0.771670 0.385835 − 0.922568i \(-0.373913\pi\)
0.385835 + 0.922568i \(0.373913\pi\)
\(30\) 0 0
\(31\) − 5.58043i − 1.00228i −0.865368 − 0.501138i \(-0.832915\pi\)
0.865368 − 0.501138i \(-0.167085\pi\)
\(32\) 0 0
\(33\) −6.25278 −1.08847
\(34\) 0 0
\(35\) −1.43739 −0.242964
\(36\) 0 0
\(37\) 3.49132 − 4.98103i 0.573970 − 0.818876i
\(38\) 0 0
\(39\) − 2.37440i − 0.380208i
\(40\) 0 0
\(41\) 5.36694 0.838175 0.419087 − 0.907946i \(-0.362350\pi\)
0.419087 + 0.907946i \(0.362350\pi\)
\(42\) 0 0
\(43\) −3.49998 −0.533742 −0.266871 − 0.963732i \(-0.585990\pi\)
−0.266871 + 0.963732i \(0.585990\pi\)
\(44\) 0 0
\(45\) −1.36027 −0.202777
\(46\) 0 0
\(47\) 2.11515 0.308526 0.154263 − 0.988030i \(-0.450700\pi\)
0.154263 + 0.988030i \(0.450700\pi\)
\(48\) 0 0
\(49\) −5.88338 −0.840484
\(50\) 0 0
\(51\) 1.36964 0.191787
\(52\) 0 0
\(53\) 0.643686i 0.0884171i 0.999022 + 0.0442085i \(0.0140766\pi\)
−0.999022 + 0.0442085i \(0.985923\pi\)
\(54\) 0 0
\(55\) − 8.50546i − 1.14688i
\(56\) 0 0
\(57\) 1.72927i 0.229048i
\(58\) 0 0
\(59\) −10.1925 −1.32694 −0.663472 − 0.748201i \(-0.730918\pi\)
−0.663472 + 0.748201i \(0.730918\pi\)
\(60\) 0 0
\(61\) −2.79008 −0.357233 −0.178616 − 0.983919i \(-0.557162\pi\)
−0.178616 + 0.983919i \(0.557162\pi\)
\(62\) 0 0
\(63\) 1.05670 0.133132
\(64\) 0 0
\(65\) 3.22981 0.400609
\(66\) 0 0
\(67\) 2.63276i 0.321642i 0.986984 + 0.160821i \(0.0514143\pi\)
−0.986984 + 0.160821i \(0.948586\pi\)
\(68\) 0 0
\(69\) −0.0334155 −0.00402275
\(70\) 0 0
\(71\) −8.36661 −0.992934 −0.496467 − 0.868056i \(-0.665370\pi\)
−0.496467 + 0.868056i \(0.665370\pi\)
\(72\) 0 0
\(73\) −4.99853 −0.585034 −0.292517 − 0.956260i \(-0.594493\pi\)
−0.292517 + 0.956260i \(0.594493\pi\)
\(74\) 0 0
\(75\) 3.14967i 0.363693i
\(76\) 0 0
\(77\) 6.60732i 0.752974i
\(78\) 0 0
\(79\) 1.86283i 0.209585i 0.994494 + 0.104793i \(0.0334179\pi\)
−0.994494 + 0.104793i \(0.966582\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) − 0.00717686i − 0.000787762i −1.00000 0.000393881i \(-0.999875\pi\)
1.00000 0.000393881i \(-0.000125376\pi\)
\(84\) 0 0
\(85\) 1.86307i 0.202078i
\(86\) 0 0
\(87\) − 4.15557i − 0.445524i
\(88\) 0 0
\(89\) − 0.818212i − 0.0867303i −0.999059 − 0.0433652i \(-0.986192\pi\)
0.999059 − 0.0433652i \(-0.0138079\pi\)
\(90\) 0 0
\(91\) −2.50903 −0.263017
\(92\) 0 0
\(93\) −5.58043 −0.578664
\(94\) 0 0
\(95\) −2.35228 −0.241338
\(96\) 0 0
\(97\) − 11.1763i − 1.13479i −0.823447 − 0.567393i \(-0.807952\pi\)
0.823447 − 0.567393i \(-0.192048\pi\)
\(98\) 0 0
\(99\) 6.25278i 0.628428i
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 3552.2.o.a.2737.7 76
4.3 odd 2 888.2.o.a.517.47 yes 76
8.3 odd 2 888.2.o.a.517.29 ✓ 76
8.5 even 2 inner 3552.2.o.a.2737.6 76
37.36 even 2 inner 3552.2.o.a.2737.5 76
148.147 odd 2 888.2.o.a.517.30 yes 76
296.147 odd 2 888.2.o.a.517.48 yes 76
296.221 even 2 inner 3552.2.o.a.2737.8 76
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
888.2.o.a.517.29 ✓ 76 8.3 odd 2
888.2.o.a.517.30 yes 76 148.147 odd 2
888.2.o.a.517.47 yes 76 4.3 odd 2
888.2.o.a.517.48 yes 76 296.147 odd 2
3552.2.o.a.2737.5 76 37.36 even 2 inner
3552.2.o.a.2737.6 76 8.5 even 2 inner
3552.2.o.a.2737.7 76 1.1 even 1 trivial
3552.2.o.a.2737.8 76 296.221 even 2 inner