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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [75,4,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, 4); 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, 4, names="a")
 
Level: \( N \) \(=\) \( 75 = 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
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: \(4.42514325043\)
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: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 3^{2} \)
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 75.4.e.a.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)
 
\(f(q)\) \(=\) \(q+(3.67423 + 3.67423i) q^{3} -8.00000i q^{4} +(22.0454 - 22.0454i) q^{7} +27.0000i q^{9} +(29.3939 - 29.3939i) q^{12} +(44.0908 + 44.0908i) q^{13} -64.0000 q^{16} -56.0000i q^{19} +162.000 q^{21} +(-99.2043 + 99.2043i) q^{27} +(-176.363 - 176.363i) q^{28} -308.000 q^{31} +216.000 q^{36} +(-308.636 + 308.636i) q^{37} +324.000i q^{39} +(154.318 + 154.318i) q^{43} +(-235.151 - 235.151i) q^{48} -629.000i q^{49} +(352.727 - 352.727i) q^{52} +(205.757 - 205.757i) q^{57} +182.000 q^{61} +(595.226 + 595.226i) q^{63} +512.000i q^{64} +(462.954 - 462.954i) q^{67} +(264.545 + 264.545i) q^{73} -448.000 q^{76} +884.000i q^{79} -729.000 q^{81} -1296.00i q^{84} +1944.00 q^{91} +(-1131.66 - 1131.66i) q^{93} +(-969.998 + 969.998i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 256 q^{16} + 648 q^{21} - 1232 q^{31} + 864 q^{36} + 728 q^{61} - 1792 q^{76} - 2916 q^{81} + 7776 q^{91}+O(q^{100}) \) 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 \(0.750000\pi\)
\(3\) 3.67423 + 3.67423i 0.707107 + 0.707107i
\(4\) − 8.00000i − 1.00000i
\(5\) 0 0
\(6\) 0 0
\(7\) 22.0454 − 22.0454i 1.19034 − 1.19034i 0.213368 − 0.976972i \(-0.431557\pi\)
0.976972 − 0.213368i \(-0.0684434\pi\)
\(8\) 0 0
\(9\) 27.0000i 1.00000i
\(10\) 0 0
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) 29.3939 − 29.3939i 0.707107 − 0.707107i
\(13\) 44.0908 + 44.0908i 0.940661 + 0.940661i 0.998335 − 0.0576745i \(-0.0183686\pi\)
−0.0576745 + 0.998335i \(0.518369\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −64.0000 −1.00000
\(17\) 0 0 0.707107 − 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(18\) 0 0
\(19\) − 56.0000i − 0.676173i −0.941115 − 0.338086i \(-0.890220\pi\)
0.941115 − 0.338086i \(-0.109780\pi\)
\(20\) 0 0
\(21\) 162.000 1.68340
\(22\) 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
\(27\) −99.2043 + 99.2043i −0.707107 + 0.707107i
\(28\) −176.363 − 176.363i −1.19034 − 1.19034i
\(29\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) 0 0
\(31\) −308.000 −1.78447 −0.892233 − 0.451576i \(-0.850862\pi\)
−0.892233 + 0.451576i \(0.850862\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 216.000 1.00000
\(37\) −308.636 + 308.636i −1.37134 + 1.37134i −0.512867 + 0.858468i \(0.671417\pi\)
−0.858468 + 0.512867i \(0.828583\pi\)
\(38\) 0 0
\(39\) 324.000i 1.33030i
\(40\) 0 0
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) 0 0
\(43\) 154.318 + 154.318i 0.547285 + 0.547285i 0.925655 − 0.378370i \(-0.123515\pi\)
−0.378370 + 0.925655i \(0.623515\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 0.707107 − 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(48\) −235.151 − 235.151i −0.707107 − 0.707107i
\(49\) − 629.000i − 1.83382i
\(50\) 0 0
\(51\) 0 0
\(52\) 352.727 − 352.727i 0.940661 − 0.940661i
\(53\) 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
\(57\) 205.757 − 205.757i 0.478126 − 0.478126i
\(58\) 0 0
\(59\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) 182.000 0.382012 0.191006 − 0.981589i \(-0.438825\pi\)
0.191006 + 0.981589i \(0.438825\pi\)
\(62\) 0 0
\(63\) 595.226 + 595.226i 1.19034 + 1.19034i
\(64\) 512.000i 1.00000i
\(65\) 0 0
\(66\) 0 0
\(67\) 462.954 − 462.954i 0.844161 − 0.844161i −0.145236 − 0.989397i \(-0.546394\pi\)
0.989397 + 0.145236i \(0.0463942\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 0 0
\(73\) 264.545 + 264.545i 0.424146 + 0.424146i 0.886628 − 0.462483i \(-0.153041\pi\)
−0.462483 + 0.886628i \(0.653041\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) −448.000 −0.676173
\(77\) 0 0
\(78\) 0 0
\(79\) 884.000i 1.25896i 0.777017 + 0.629480i \(0.216732\pi\)
−0.777017 + 0.629480i \(0.783268\pi\)
\(80\) 0 0
\(81\) −729.000 −1.00000
\(82\) 0 0
\(83\) 0 0 −0.707107 − 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(84\) − 1296.00i − 1.68340i
\(85\) 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
\(91\) 1944.00 2.23941
\(92\) 0 0
\(93\) −1131.66 − 1131.66i −1.26181 − 1.26181i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −969.998 + 969.998i −1.01534 + 1.01534i −0.0154636 + 0.999880i \(0.504922\pi\)
−0.999880 + 0.0154636i \(0.995078\pi\)
\(98\) 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 75.4.e.a.68.2 yes 4
3.2 odd 2 CM 75.4.e.a.68.2 yes 4
5.2 odd 4 inner 75.4.e.a.32.2 yes 4
5.3 odd 4 inner 75.4.e.a.32.1 ✓ 4
5.4 even 2 inner 75.4.e.a.68.1 yes 4
15.2 even 4 inner 75.4.e.a.32.2 yes 4
15.8 even 4 inner 75.4.e.a.32.1 ✓ 4
15.14 odd 2 inner 75.4.e.a.68.1 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
75.4.e.a.32.1 ✓ 4 5.3 odd 4 inner
75.4.e.a.32.1 ✓ 4 15.8 even 4 inner
75.4.e.a.32.2 yes 4 5.2 odd 4 inner
75.4.e.a.32.2 yes 4 15.2 even 4 inner
75.4.e.a.68.1 yes 4 5.4 even 2 inner
75.4.e.a.68.1 yes 4 15.14 odd 2 inner
75.4.e.a.68.2 yes 4 1.1 even 1 trivial
75.4.e.a.68.2 yes 4 3.2 odd 2 CM