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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [675,3,Mod(674,675)] 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("675.674"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(675, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 675 = 3^{3} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 675.d (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,-6,0,10,0,0,0,-6,0,0,0,0,0,0,0,-22,-36,0,32] 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: \(18.3924178443\)
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: \( 5 \)
Twist minimal: no (minimal twist has level 27)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 674.2
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 675.674
Dual form 675.3.d.a.674.1

$q$-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)
 
\(f(q)\) \(=\) \(q-3.00000 q^{2} +5.00000 q^{4} +5.00000i q^{7} -3.00000 q^{8} -15.0000i q^{11} +10.0000i q^{13} -15.0000i q^{14} -11.0000 q^{16} -18.0000 q^{17} +16.0000 q^{19} +45.0000i q^{22} -12.0000 q^{23} -30.0000i q^{26} +25.0000i q^{28} -30.0000i q^{29} -1.00000 q^{31} +45.0000 q^{32} +54.0000 q^{34} +20.0000i q^{37} -48.0000 q^{38} +60.0000i q^{41} -50.0000i q^{43} -75.0000i q^{44} +36.0000 q^{46} +6.00000 q^{47} +24.0000 q^{49} +50.0000i q^{52} -27.0000 q^{53} -15.0000i q^{56} +90.0000i q^{58} +30.0000i q^{59} -76.0000 q^{61} +3.00000 q^{62} -91.0000 q^{64} -10.0000i q^{67} -90.0000 q^{68} -90.0000i q^{71} -65.0000i q^{73} -60.0000i q^{74} +80.0000 q^{76} +75.0000 q^{77} -14.0000 q^{79} -180.000i q^{82} +3.00000 q^{83} +150.000i q^{86} +45.0000i q^{88} -90.0000i q^{89} -50.0000 q^{91} -60.0000 q^{92} -18.0000 q^{94} -85.0000i q^{97} -72.0000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 6 q^{2} + 10 q^{4} - 6 q^{8} - 22 q^{16} - 36 q^{17} + 32 q^{19} - 24 q^{23} - 2 q^{31} + 90 q^{32} + 108 q^{34} - 96 q^{38} + 72 q^{46} + 12 q^{47} + 48 q^{49} - 54 q^{53} - 152 q^{61} + 6 q^{62}+ \cdots - 144 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/675\mathbb{Z}\right)^\times\).

\(n\) \(326\) \(352\)
\(\chi(n)\) \(-1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −3.00000 −1.50000 −0.750000 − 0.661438i \(-0.769947\pi\)
−0.750000 + 0.661438i \(0.769947\pi\)
\(3\) 0 0
\(4\) 5.00000 1.25000
\(5\) 0 0
\(6\) 0 0
\(7\) 5.00000i 0.714286i 0.934050 + 0.357143i \(0.116249\pi\)
−0.934050 + 0.357143i \(0.883751\pi\)
\(8\) −3.00000 −0.375000
\(9\) 0 0
\(10\) 0 0
\(11\) − 15.0000i − 1.36364i −0.731522 − 0.681818i \(-0.761190\pi\)
0.731522 − 0.681818i \(-0.238810\pi\)
\(12\) 0 0
\(13\) 10.0000i 0.769231i 0.923077 + 0.384615i \(0.125666\pi\)
−0.923077 + 0.384615i \(0.874334\pi\)
\(14\) − 15.0000i − 1.07143i
\(15\) 0 0
\(16\) −11.0000 −0.687500
\(17\) −18.0000 −1.05882 −0.529412 − 0.848365i \(-0.677587\pi\)
−0.529412 + 0.848365i \(0.677587\pi\)
\(18\) 0 0
\(19\) 16.0000 0.842105 0.421053 − 0.907036i \(-0.361661\pi\)
0.421053 + 0.907036i \(0.361661\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 45.0000i 2.04545i
\(23\) −12.0000 −0.521739 −0.260870 − 0.965374i \(-0.584009\pi\)
−0.260870 + 0.965374i \(0.584009\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) − 30.0000i − 1.15385i
\(27\) 0 0
\(28\) 25.0000i 0.892857i
\(29\) − 30.0000i − 1.03448i −0.855840 − 0.517241i \(-0.826959\pi\)
0.855840 − 0.517241i \(-0.173041\pi\)
\(30\) 0 0
\(31\) −1.00000 −0.0322581 −0.0161290 − 0.999870i \(-0.505134\pi\)
−0.0161290 + 0.999870i \(0.505134\pi\)
\(32\) 45.0000 1.40625
\(33\) 0 0
\(34\) 54.0000 1.58824
\(35\) 0 0
\(36\) 0 0
\(37\) 20.0000i 0.540541i 0.962784 + 0.270270i \(0.0871131\pi\)
−0.962784 + 0.270270i \(0.912887\pi\)
\(38\) −48.0000 −1.26316
\(39\) 0 0
\(40\) 0 0
\(41\) 60.0000i 1.46341i 0.681619 + 0.731707i \(0.261276\pi\)
−0.681619 + 0.731707i \(0.738724\pi\)
\(42\) 0 0
\(43\) − 50.0000i − 1.16279i −0.813621 − 0.581395i \(-0.802507\pi\)
0.813621 − 0.581395i \(-0.197493\pi\)
\(44\) − 75.0000i − 1.70455i
\(45\) 0 0
\(46\) 36.0000 0.782609
\(47\) 6.00000 0.127660 0.0638298 − 0.997961i \(-0.479669\pi\)
0.0638298 + 0.997961i \(0.479669\pi\)
\(48\) 0 0
\(49\) 24.0000 0.489796
\(50\) 0 0
\(51\) 0 0
\(52\) 50.0000i 0.961538i
\(53\) −27.0000 −0.509434 −0.254717 − 0.967016i \(-0.581982\pi\)
−0.254717 + 0.967016i \(0.581982\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) − 15.0000i − 0.267857i
\(57\) 0 0
\(58\) 90.0000i 1.55172i
\(59\) 30.0000i 0.508475i 0.967142 + 0.254237i \(0.0818244\pi\)
−0.967142 + 0.254237i \(0.918176\pi\)
\(60\) 0 0
\(61\) −76.0000 −1.24590 −0.622951 − 0.782261i \(-0.714066\pi\)
−0.622951 + 0.782261i \(0.714066\pi\)
\(62\) 3.00000 0.0483871
\(63\) 0 0
\(64\) −91.0000 −1.42188
\(65\) 0 0
\(66\) 0 0
\(67\) − 10.0000i − 0.149254i −0.997212 − 0.0746269i \(-0.976223\pi\)
0.997212 − 0.0746269i \(-0.0237766\pi\)
\(68\) −90.0000 −1.32353
\(69\) 0 0
\(70\) 0 0
\(71\) − 90.0000i − 1.26761i −0.773495 − 0.633803i \(-0.781493\pi\)
0.773495 − 0.633803i \(-0.218507\pi\)
\(72\) 0 0
\(73\) − 65.0000i − 0.890411i −0.895428 − 0.445205i \(-0.853131\pi\)
0.895428 − 0.445205i \(-0.146869\pi\)
\(74\) − 60.0000i − 0.810811i
\(75\) 0 0
\(76\) 80.0000 1.05263
\(77\) 75.0000 0.974026
\(78\) 0 0
\(79\) −14.0000 −0.177215 −0.0886076 − 0.996067i \(-0.528242\pi\)
−0.0886076 + 0.996067i \(0.528242\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) − 180.000i − 2.19512i
\(83\) 3.00000 0.0361446 0.0180723 − 0.999837i \(-0.494247\pi\)
0.0180723 + 0.999837i \(0.494247\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 150.000i 1.74419i
\(87\) 0 0
\(88\) 45.0000i 0.511364i
\(89\) − 90.0000i − 1.01124i −0.862757 − 0.505618i \(-0.831265\pi\)
0.862757 − 0.505618i \(-0.168735\pi\)
\(90\) 0 0
\(91\) −50.0000 −0.549451
\(92\) −60.0000 −0.652174
\(93\) 0 0
\(94\) −18.0000 −0.191489
\(95\) 0 0
\(96\) 0 0
\(97\) − 85.0000i − 0.876289i −0.898905 − 0.438144i \(-0.855636\pi\)
0.898905 − 0.438144i \(-0.144364\pi\)
\(98\) −72.0000 −0.734694
\(99\) 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 675.3.d.a.674.2 2
3.2 odd 2 675.3.d.d.674.2 2
5.2 odd 4 675.3.c.h.26.1 2
5.3 odd 4 27.3.b.b.26.2 yes 2
5.4 even 2 675.3.d.d.674.1 2
15.2 even 4 675.3.c.h.26.2 2
15.8 even 4 27.3.b.b.26.1 ✓ 2
15.14 odd 2 inner 675.3.d.a.674.1 2
20.3 even 4 432.3.e.c.161.1 2
40.3 even 4 1728.3.e.g.1025.2 2
40.13 odd 4 1728.3.e.m.1025.2 2
45.13 odd 12 81.3.d.b.26.1 4
45.23 even 12 81.3.d.b.26.2 4
45.38 even 12 81.3.d.b.53.1 4
45.43 odd 12 81.3.d.b.53.2 4
60.23 odd 4 432.3.e.c.161.2 2
120.53 even 4 1728.3.e.m.1025.1 2
120.83 odd 4 1728.3.e.g.1025.1 2
180.23 odd 12 1296.3.q.j.593.2 4
180.43 even 12 1296.3.q.j.1025.2 4
180.83 odd 12 1296.3.q.j.1025.1 4
180.103 even 12 1296.3.q.j.593.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
27.3.b.b.26.1 ✓ 2 15.8 even 4
27.3.b.b.26.2 yes 2 5.3 odd 4
81.3.d.b.26.1 4 45.13 odd 12
81.3.d.b.26.2 4 45.23 even 12
81.3.d.b.53.1 4 45.38 even 12
81.3.d.b.53.2 4 45.43 odd 12
432.3.e.c.161.1 2 20.3 even 4
432.3.e.c.161.2 2 60.23 odd 4
675.3.c.h.26.1 2 5.2 odd 4
675.3.c.h.26.2 2 15.2 even 4
675.3.d.a.674.1 2 15.14 odd 2 inner
675.3.d.a.674.2 2 1.1 even 1 trivial
675.3.d.d.674.1 2 5.4 even 2
675.3.d.d.674.2 2 3.2 odd 2
1296.3.q.j.593.1 4 180.103 even 12
1296.3.q.j.593.2 4 180.23 odd 12
1296.3.q.j.1025.1 4 180.83 odd 12
1296.3.q.j.1025.2 4 180.43 even 12
1728.3.e.g.1025.1 2 120.83 odd 4
1728.3.e.g.1025.2 2 40.3 even 4
1728.3.e.m.1025.1 2 120.53 even 4
1728.3.e.m.1025.2 2 40.13 odd 4