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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [75,12,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, 12); 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, 12, names="a")
 
Level: \( N \) \(=\) \( 75 = 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 12 \)
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: \(57.6257385420\)
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: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{4}]$

Embedding invariants

Embedding label 68.1
Root \(-1.22474 + 1.22474i\) of defining polynomial
Character \(\chi\) \(=\) 75.68
Dual form 75.12.e.a.32.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-297.613 - 297.613i) q^{3} -2048.00i q^{4} +(31607.0 - 31607.0i) q^{7} +177147. i q^{9} +(-609511. + 609511. i) q^{12} +(-1.02771e6 - 1.02771e6i) q^{13} -4.19430e6 q^{16} -2.05819e7i q^{19} -1.88133e7 q^{21} +(5.27213e7 - 5.27213e7i) q^{27} +(-6.47311e7 - 6.47311e7i) q^{28} -2.96477e8 q^{31} +3.62797e8 q^{36} +(2.22157e8 - 2.22157e8i) q^{37} +6.11719e8i q^{39} +(5.43178e8 + 5.43178e8i) q^{43} +(1.24828e9 + 1.24828e9i) q^{48} -2.06770e7i q^{49} +(-2.10475e9 + 2.10475e9i) q^{52} +(-6.12544e9 + 6.12544e9i) q^{57} -1.29773e10 q^{61} +(5.59908e9 + 5.59908e9i) q^{63} +8.58993e9i q^{64} +(1.50516e10 - 1.50516e10i) q^{67} +(2.07683e10 + 2.07683e10i) q^{73} -4.21517e10 q^{76} +3.28858e10i q^{79} -3.13811e10 q^{81} +3.85296e10i q^{84} -6.49656e10 q^{91} +(8.82354e10 + 8.82354e10i) q^{93} +(-1.13592e11 + 1.13592e11i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 16777216 q^{16} - 75253212 q^{21} - 1185907772 q^{31} + 1451188224 q^{36} - 51909171652 q^{61} - 168606932992 q^{76} - 125524238436 q^{81} - 259862347764 q^{91}+O(q^{100}) \) 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 \(0.750000\pi\)
\(3\) −297.613 − 297.613i −0.707107 − 0.707107i
\(4\) − 2048.00i − 1.00000i
\(5\) 0 0
\(6\) 0 0
\(7\) 31607.0 − 31607.0i 0.710794 − 0.710794i −0.255907 − 0.966701i \(-0.582374\pi\)
0.966701 + 0.255907i \(0.0823741\pi\)
\(8\) 0 0
\(9\) 177147.i 1.00000i
\(10\) 0 0
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) −609511. + 609511.i −0.707107 + 0.707107i
\(13\) −1.02771e6 − 1.02771e6i −0.767683 − 0.767683i 0.210015 − 0.977698i \(-0.432649\pi\)
−0.977698 + 0.210015i \(0.932649\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −4.19430e6 −1.00000
\(17\) 0 0 0.707107 − 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(18\) 0 0
\(19\) − 2.05819e7i − 1.90696i −0.301463 − 0.953478i \(-0.597475\pi\)
0.301463 − 0.953478i \(-0.402525\pi\)
\(20\) 0 0
\(21\) −1.88133e7 −1.00521
\(22\) 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
\(27\) 5.27213e7 − 5.27213e7i 0.707107 − 0.707107i
\(28\) −6.47311e7 − 6.47311e7i −0.710794 − 0.710794i
\(29\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) 0 0
\(31\) −2.96477e8 −1.85995 −0.929976 − 0.367621i \(-0.880172\pi\)
−0.929976 + 0.367621i \(0.880172\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 3.62797e8 1.00000
\(37\) 2.22157e8 − 2.22157e8i 0.526684 − 0.526684i −0.392898 − 0.919582i \(-0.628528\pi\)
0.919582 + 0.392898i \(0.128528\pi\)
\(38\) 0 0
\(39\) 6.11719e8i 1.08567i
\(40\) 0 0
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) 0 0
\(43\) 5.43178e8 + 5.43178e8i 0.563463 + 0.563463i 0.930289 − 0.366826i \(-0.119556\pi\)
−0.366826 + 0.930289i \(0.619556\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 0.707107 − 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(48\) 1.24828e9 + 1.24828e9i 0.707107 + 0.707107i
\(49\) − 2.06770e7i − 0.0104570i
\(50\) 0 0
\(51\) 0 0
\(52\) −2.10475e9 + 2.10475e9i −0.767683 + 0.767683i
\(53\) 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
\(57\) −6.12544e9 + 6.12544e9i −1.34842 + 1.34842i
\(58\) 0 0
\(59\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) −1.29773e10 −1.96730 −0.983649 − 0.180098i \(-0.942359\pi\)
−0.983649 + 0.180098i \(0.942359\pi\)
\(62\) 0 0
\(63\) 5.59908e9 + 5.59908e9i 0.710794 + 0.710794i
\(64\) 8.58993e9i 1.00000i
\(65\) 0 0
\(66\) 0 0
\(67\) 1.50516e10 − 1.50516e10i 1.36198 − 1.36198i 0.490597 − 0.871387i \(-0.336779\pi\)
0.871387 − 0.490597i \(-0.163221\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 0 0
\(73\) 2.07683e10 + 2.07683e10i 1.17254 + 1.17254i 0.981603 + 0.190933i \(0.0611512\pi\)
0.190933 + 0.981603i \(0.438849\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) −4.21517e10 −1.90696
\(77\) 0 0
\(78\) 0 0
\(79\) 3.28858e10i 1.20243i 0.799087 + 0.601215i \(0.205316\pi\)
−0.799087 + 0.601215i \(0.794684\pi\)
\(80\) 0 0
\(81\) −3.13811e10 −1.00000
\(82\) 0 0
\(83\) 0 0 −0.707107 − 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(84\) 3.85296e10i 1.00521i
\(85\) 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
\(91\) −6.49656e10 −1.09133
\(92\) 0 0
\(93\) 8.82354e10 + 8.82354e10i 1.31518 + 1.31518i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −1.13592e11 + 1.13592e11i −1.34309 + 1.34309i −0.450117 + 0.892969i \(0.648618\pi\)
−0.892969 + 0.450117i \(0.851382\pi\)
\(98\) 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 75.12.e.a.68.1 yes 4
3.2 odd 2 CM 75.12.e.a.68.1 yes 4
5.2 odd 4 inner 75.12.e.a.32.1 ✓ 4
5.3 odd 4 inner 75.12.e.a.32.2 yes 4
5.4 even 2 inner 75.12.e.a.68.2 yes 4
15.2 even 4 inner 75.12.e.a.32.1 ✓ 4
15.8 even 4 inner 75.12.e.a.32.2 yes 4
15.14 odd 2 inner 75.12.e.a.68.2 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
75.12.e.a.32.1 ✓ 4 5.2 odd 4 inner
75.12.e.a.32.1 ✓ 4 15.2 even 4 inner
75.12.e.a.32.2 yes 4 5.3 odd 4 inner
75.12.e.a.32.2 yes 4 15.8 even 4 inner
75.12.e.a.68.1 yes 4 1.1 even 1 trivial
75.12.e.a.68.1 yes 4 3.2 odd 2 CM
75.12.e.a.68.2 yes 4 5.4 even 2 inner
75.12.e.a.68.2 yes 4 15.14 odd 2 inner