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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [675,3,Mod(26,675)] 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("675.26"); 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, 0])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 675 = 3^{3} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 675.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,-10,0,0,-10] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == 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: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 27)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 26.1
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 675.26
Dual form 675.3.c.h.26.2

$q$-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)
 
\(f(q)\) \(=\) \(q-3.00000i q^{2} -5.00000 q^{4} -5.00000 q^{7} +3.00000i q^{8} -15.0000i q^{11} +10.0000 q^{13} +15.0000i q^{14} -11.0000 q^{16} -18.0000i q^{17} -16.0000 q^{19} -45.0000 q^{22} +12.0000i q^{23} -30.0000i q^{26} +25.0000 q^{28} +30.0000i q^{29} -1.00000 q^{31} +45.0000i q^{32} -54.0000 q^{34} -20.0000 q^{37} +48.0000i q^{38} +60.0000i q^{41} -50.0000 q^{43} +75.0000i q^{44} +36.0000 q^{46} +6.00000i q^{47} -24.0000 q^{49} -50.0000 q^{52} +27.0000i q^{53} -15.0000i q^{56} +90.0000 q^{58} -30.0000i q^{59} -76.0000 q^{61} +3.00000i q^{62} +91.0000 q^{64} +10.0000 q^{67} +90.0000i q^{68} -90.0000i q^{71} -65.0000 q^{73} +60.0000i q^{74} +80.0000 q^{76} +75.0000i q^{77} +14.0000 q^{79} +180.000 q^{82} -3.00000i q^{83} +150.000i q^{86} +45.0000 q^{88} +90.0000i q^{89} -50.0000 q^{91} -60.0000i q^{92} +18.0000 q^{94} +85.0000 q^{97} +72.0000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 10 q^{4} - 10 q^{7} + 20 q^{13} - 22 q^{16} - 32 q^{19} - 90 q^{22} + 50 q^{28} - 2 q^{31} - 108 q^{34} - 40 q^{37} - 100 q^{43} + 72 q^{46} - 48 q^{49} - 100 q^{52} + 180 q^{58} - 152 q^{61} + 182 q^{64}+ \cdots + 170 q^{97}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 3.00000i − 1.50000i −0.661438 − 0.750000i \(-0.730053\pi\)
0.661438 − 0.750000i \(-0.269947\pi\)
\(3\) 0 0
\(4\) −5.00000 −1.25000
\(5\) 0 0
\(6\) 0 0
\(7\) −5.00000 −0.714286 −0.357143 − 0.934050i \(-0.616249\pi\)
−0.357143 + 0.934050i \(0.616249\pi\)
\(8\) 3.00000i 0.375000i
\(9\) 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
\(13\) 10.0000 0.769231 0.384615 − 0.923077i \(-0.374334\pi\)
0.384615 + 0.923077i \(0.374334\pi\)
\(14\) 15.0000i 1.07143i
\(15\) 0 0
\(16\) −11.0000 −0.687500
\(17\) − 18.0000i − 1.05882i −0.848365 − 0.529412i \(-0.822413\pi\)
0.848365 − 0.529412i \(-0.177587\pi\)
\(18\) 0 0
\(19\) −16.0000 −0.842105 −0.421053 − 0.907036i \(-0.638339\pi\)
−0.421053 + 0.907036i \(0.638339\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −45.0000 −2.04545
\(23\) 12.0000i 0.521739i 0.965374 + 0.260870i \(0.0840093\pi\)
−0.965374 + 0.260870i \(0.915991\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) − 30.0000i − 1.15385i
\(27\) 0 0
\(28\) 25.0000 0.892857
\(29\) 30.0000i 1.03448i 0.855840 + 0.517241i \(0.173041\pi\)
−0.855840 + 0.517241i \(0.826959\pi\)
\(30\) 0 0
\(31\) −1.00000 −0.0322581 −0.0161290 − 0.999870i \(-0.505134\pi\)
−0.0161290 + 0.999870i \(0.505134\pi\)
\(32\) 45.0000i 1.40625i
\(33\) 0 0
\(34\) −54.0000 −1.58824
\(35\) 0 0
\(36\) 0 0
\(37\) −20.0000 −0.540541 −0.270270 − 0.962784i \(-0.587113\pi\)
−0.270270 + 0.962784i \(0.587113\pi\)
\(38\) 48.0000i 1.26316i
\(39\) 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
\(43\) −50.0000 −1.16279 −0.581395 − 0.813621i \(-0.697493\pi\)
−0.581395 + 0.813621i \(0.697493\pi\)
\(44\) 75.0000i 1.70455i
\(45\) 0 0
\(46\) 36.0000 0.782609
\(47\) 6.00000i 0.127660i 0.997961 + 0.0638298i \(0.0203315\pi\)
−0.997961 + 0.0638298i \(0.979669\pi\)
\(48\) 0 0
\(49\) −24.0000 −0.489796
\(50\) 0 0
\(51\) 0 0
\(52\) −50.0000 −0.961538
\(53\) 27.0000i 0.509434i 0.967016 + 0.254717i \(0.0819823\pi\)
−0.967016 + 0.254717i \(0.918018\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) − 15.0000i − 0.267857i
\(57\) 0 0
\(58\) 90.0000 1.55172
\(59\) − 30.0000i − 0.508475i −0.967142 − 0.254237i \(-0.918176\pi\)
0.967142 − 0.254237i \(-0.0818244\pi\)
\(60\) 0 0
\(61\) −76.0000 −1.24590 −0.622951 − 0.782261i \(-0.714066\pi\)
−0.622951 + 0.782261i \(0.714066\pi\)
\(62\) 3.00000i 0.0483871i
\(63\) 0 0
\(64\) 91.0000 1.42188
\(65\) 0 0
\(66\) 0 0
\(67\) 10.0000 0.149254 0.0746269 − 0.997212i \(-0.476223\pi\)
0.0746269 + 0.997212i \(0.476223\pi\)
\(68\) 90.0000i 1.32353i
\(69\) 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
\(73\) −65.0000 −0.890411 −0.445205 − 0.895428i \(-0.646869\pi\)
−0.445205 + 0.895428i \(0.646869\pi\)
\(74\) 60.0000i 0.810811i
\(75\) 0 0
\(76\) 80.0000 1.05263
\(77\) 75.0000i 0.974026i
\(78\) 0 0
\(79\) 14.0000 0.177215 0.0886076 − 0.996067i \(-0.471758\pi\)
0.0886076 + 0.996067i \(0.471758\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 180.000 2.19512
\(83\) − 3.00000i − 0.0361446i −0.999837 − 0.0180723i \(-0.994247\pi\)
0.999837 − 0.0180723i \(-0.00575290\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 150.000i 1.74419i
\(87\) 0 0
\(88\) 45.0000 0.511364
\(89\) 90.0000i 1.01124i 0.862757 + 0.505618i \(0.168735\pi\)
−0.862757 + 0.505618i \(0.831265\pi\)
\(90\) 0 0
\(91\) −50.0000 −0.549451
\(92\) − 60.0000i − 0.652174i
\(93\) 0 0
\(94\) 18.0000 0.191489
\(95\) 0 0
\(96\) 0 0
\(97\) 85.0000 0.876289 0.438144 − 0.898905i \(-0.355636\pi\)
0.438144 + 0.898905i \(0.355636\pi\)
\(98\) 72.0000i 0.734694i
\(99\) 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.c.h.26.1 2
3.2 odd 2 inner 675.3.c.h.26.2 2
5.2 odd 4 675.3.d.d.674.1 2
5.3 odd 4 675.3.d.a.674.2 2
5.4 even 2 27.3.b.b.26.2 yes 2
15.2 even 4 675.3.d.a.674.1 2
15.8 even 4 675.3.d.d.674.2 2
15.14 odd 2 27.3.b.b.26.1 ✓ 2
20.19 odd 2 432.3.e.c.161.1 2
40.19 odd 2 1728.3.e.g.1025.2 2
40.29 even 2 1728.3.e.m.1025.2 2
45.4 even 6 81.3.d.b.26.1 4
45.14 odd 6 81.3.d.b.26.2 4
45.29 odd 6 81.3.d.b.53.1 4
45.34 even 6 81.3.d.b.53.2 4
60.59 even 2 432.3.e.c.161.2 2
120.29 odd 2 1728.3.e.m.1025.1 2
120.59 even 2 1728.3.e.g.1025.1 2
180.59 even 6 1296.3.q.j.593.2 4
180.79 odd 6 1296.3.q.j.1025.2 4
180.119 even 6 1296.3.q.j.1025.1 4
180.139 odd 6 1296.3.q.j.593.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
27.3.b.b.26.1 ✓ 2 15.14 odd 2
27.3.b.b.26.2 yes 2 5.4 even 2
81.3.d.b.26.1 4 45.4 even 6
81.3.d.b.26.2 4 45.14 odd 6
81.3.d.b.53.1 4 45.29 odd 6
81.3.d.b.53.2 4 45.34 even 6
432.3.e.c.161.1 2 20.19 odd 2
432.3.e.c.161.2 2 60.59 even 2
675.3.c.h.26.1 2 1.1 even 1 trivial
675.3.c.h.26.2 2 3.2 odd 2 inner
675.3.d.a.674.1 2 15.2 even 4
675.3.d.a.674.2 2 5.3 odd 4
675.3.d.d.674.1 2 5.2 odd 4
675.3.d.d.674.2 2 15.8 even 4
1296.3.q.j.593.1 4 180.139 odd 6
1296.3.q.j.593.2 4 180.59 even 6
1296.3.q.j.1025.1 4 180.119 even 6
1296.3.q.j.1025.2 4 180.79 odd 6
1728.3.e.g.1025.1 2 120.59 even 2
1728.3.e.g.1025.2 2 40.19 odd 2
1728.3.e.m.1025.1 2 120.29 odd 2
1728.3.e.m.1025.2 2 40.29 even 2