Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [75,10,Mod(32,75)] 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("75.32"); S:= CuspForms(chi, 10); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(75, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([2, 1])) N = Newforms(chi, 10, names="a")
 
Level: \( N \) \(=\) \( 75 = 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 75.e (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(6)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(38.6276877123\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(i, \sqrt{6})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3^{6} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{4}]$

Embedding invariants

Embedding label 68.2
Root \(1.22474 - 1.22474i\) of defining polynomial
Character \(\chi\) \(=\) 75.68
Dual form 75.10.e.b.32.2

$q$-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)
 
\(f(q)\) \(=\) \(q+(99.2043 + 99.2043i) q^{3} -512.000i q^{4} +(-1256.59 + 1256.59i) q^{7} +19683.0i q^{9} +(50792.6 - 50792.6i) q^{12} +(-119177. - 119177. i) q^{13} -262144. q^{16} +976696. i q^{19} -249318. q^{21} +(-1.95264e6 + 1.95264e6i) q^{27} +(643373. + 643373. i) q^{28} -1.69123e6 q^{31} +1.00777e7 q^{36} +(-1.18988e7 + 1.18988e7i) q^{37} -2.36458e7i q^{39} +(-2.94582e7 - 2.94582e7i) q^{43} +(-2.60058e7 - 2.60058e7i) q^{48} +3.71956e7i q^{49} +(-6.10189e7 + 6.10189e7i) q^{52} +(-9.68925e7 + 9.68925e7i) q^{57} -1.17903e8 q^{61} +(-2.47334e7 - 2.47334e7i) q^{63} +1.34218e8i q^{64} +(-2.19272e8 + 2.19272e8i) q^{67} +(-2.71710e8 - 2.71710e8i) q^{73} +5.00068e8 q^{76} -6.16732e8i q^{79} -3.87420e8 q^{81} +1.27651e8i q^{84} +2.99514e8 q^{91} +(-1.67777e8 - 1.67777e8i) q^{93} +(8.30538e8 - 8.30538e8i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 1048576 q^{16} - 997272 q^{21} - 6764912 q^{31} + 40310784 q^{36} - 471612232 q^{61} + 2000273408 q^{76} - 1549681956 q^{81} + 1198056096 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/75\mathbb{Z}\right)^\times\).

\(n\) \(26\) \(52\)
\(\chi(n)\) \(-1\) \(e\left(\frac{3}{4}\right)\)

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 0.707107 − 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(3\) 99.2043 + 99.2043i 0.707107 + 0.707107i
\(4\) − 512.000i − 1.00000i
\(5\) 0 0
\(6\) 0 0
\(7\) −1256.59 + 1256.59i −0.197812 + 0.197812i −0.799061 − 0.601250i \(-0.794670\pi\)
0.601250 + 0.799061i \(0.294670\pi\)
\(8\) 0 0
\(9\) 19683.0i 1.00000i
\(10\) 0 0
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) 50792.6 − 50792.6i 0.707107 − 0.707107i
\(13\) −119177. − 119177.i −1.15731 − 1.15731i −0.985052 − 0.172256i \(-0.944894\pi\)
−0.172256 − 0.985052i \(-0.555106\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −262144. −1.00000
\(17\) 0 0 0.707107 − 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(18\) 0 0
\(19\) 976696.i 1.71937i 0.510828 + 0.859683i \(0.329339\pi\)
−0.510828 + 0.859683i \(0.670661\pi\)
\(20\) 0 0
\(21\) −249318. −0.279748
\(22\) 0 0
\(23\) 0 0 −0.707107 − 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) −1.95264e6 + 1.95264e6i −0.707107 + 0.707107i
\(28\) 643373. + 643373.i 0.197812 + 0.197812i
\(29\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) 0 0
\(31\) −1.69123e6 −0.328908 −0.164454 − 0.986385i \(-0.552586\pi\)
−0.164454 + 0.986385i \(0.552586\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 1.00777e7 1.00000
\(37\) −1.18988e7 + 1.18988e7i −1.04375 + 1.04375i −0.0447521 + 0.998998i \(0.514250\pi\)
−0.998998 + 0.0447521i \(0.985750\pi\)
\(38\) 0 0
\(39\) − 2.36458e7i − 1.63668i
\(40\) 0 0
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) 0 0
\(43\) −2.94582e7 − 2.94582e7i −1.31401 − 1.31401i −0.918434 − 0.395574i \(-0.870546\pi\)
−0.395574 − 0.918434i \(-0.629454\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 0.707107 − 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(48\) −2.60058e7 − 2.60058e7i −0.707107 − 0.707107i
\(49\) 3.71956e7i 0.921741i
\(50\) 0 0
\(51\) 0 0
\(52\) −6.10189e7 + 6.10189e7i −1.15731 + 1.15731i
\(53\) 0 0 −0.707107 − 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −9.68925e7 + 9.68925e7i −1.21577 + 1.21577i
\(58\) 0 0
\(59\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) −1.17903e8 −1.09029 −0.545143 − 0.838343i \(-0.683525\pi\)
−0.545143 + 0.838343i \(0.683525\pi\)
\(62\) 0 0
\(63\) −2.47334e7 − 2.47334e7i −0.197812 − 0.197812i
\(64\) 1.34218e8i 1.00000i
\(65\) 0 0
\(66\) 0 0
\(67\) −2.19272e8 + 2.19272e8i −1.32937 + 1.32937i −0.423454 + 0.905917i \(0.639183\pi\)
−0.905917 + 0.423454i \(0.860817\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 0 0
\(73\) −2.71710e8 − 2.71710e8i −1.11983 − 1.11983i −0.991766 − 0.128064i \(-0.959124\pi\)
−0.128064 − 0.991766i \(-0.540876\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 5.00068e8 1.71937
\(77\) 0 0
\(78\) 0 0
\(79\) − 6.16732e8i − 1.78145i −0.454538 − 0.890727i \(-0.650196\pi\)
0.454538 − 0.890727i \(-0.349804\pi\)
\(80\) 0 0
\(81\) −3.87420e8 −1.00000
\(82\) 0 0
\(83\) 0 0 −0.707107 − 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(84\) 1.27651e8i 0.279748i
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) 0 0
\(91\) 2.99514e8 0.457858
\(92\) 0 0
\(93\) −1.67777e8 − 1.67777e8i −0.232573 − 0.232573i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 8.30538e8 − 8.30538e8i 0.952548 − 0.952548i −0.0463759 − 0.998924i \(-0.514767\pi\)
0.998924 + 0.0463759i \(0.0147672\pi\)
\(98\) 0 0
\(99\) 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 75.10.e.b.68.2 yes 4
3.2 odd 2 CM 75.10.e.b.68.2 yes 4
5.2 odd 4 inner 75.10.e.b.32.2 yes 4
5.3 odd 4 inner 75.10.e.b.32.1 ✓ 4
5.4 even 2 inner 75.10.e.b.68.1 yes 4
15.2 even 4 inner 75.10.e.b.32.2 yes 4
15.8 even 4 inner 75.10.e.b.32.1 ✓ 4
15.14 odd 2 inner 75.10.e.b.68.1 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
75.10.e.b.32.1 ✓ 4 5.3 odd 4 inner
75.10.e.b.32.1 ✓ 4 15.8 even 4 inner
75.10.e.b.32.2 yes 4 5.2 odd 4 inner
75.10.e.b.32.2 yes 4 15.2 even 4 inner
75.10.e.b.68.1 yes 4 5.4 even 2 inner
75.10.e.b.68.1 yes 4 15.14 odd 2 inner
75.10.e.b.68.2 yes 4 1.1 even 1 trivial
75.10.e.b.68.2 yes 4 3.2 odd 2 CM