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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [75,8,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, 8); 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, 8, names="a")
 
Level: \( N \) \(=\) \( 75 = 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 8 \)
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: \(23.4288769113\)
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 32.2
Root \(1.22474 + 1.22474i\) of defining polynomial
Character \(\chi\) \(=\) 75.32
Dual form 75.8.e.b.68.2

$q$-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)
 
\(f(q)\) \(=\) \(q+(33.0681 - 33.0681i) q^{3} +128.000i q^{4} +(927.132 + 927.132i) q^{7} -2187.00i q^{9} +(4232.72 + 4232.72i) q^{12} +(-6786.31 + 6786.31i) q^{13} -16384.0 q^{16} +43091.0i q^{19} +61317.0 q^{21} +(-72320.0 - 72320.0i) q^{27} +(-118673. + 118673. i) q^{28} +331387. q^{31} +279936. q^{36} +(388259. + 388259. i) q^{37} +448821. i q^{39} +(-678081. + 678081. i) q^{43} +(-541788. + 541788. i) q^{48} +895604. i q^{49} +(-868648. - 868648. i) q^{52} +(1.42494e6 + 1.42494e6i) q^{57} +1.99835e6 q^{61} +(2.02764e6 - 2.02764e6i) q^{63} -2.09715e6i q^{64} +(-1.97127e6 - 1.97127e6i) q^{67} +(1.55183e6 - 1.55183e6i) q^{73} -5.51565e6 q^{76} -8.76304e6i q^{79} -4.78297e6 q^{81} +7.84858e6i q^{84} -1.25836e7 q^{91} +(1.09583e7 - 1.09583e7i) q^{93} +(2.84502e6 + 2.84502e6i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 65536 q^{16} + 245268 q^{21} + 1325548 q^{31} + 1119744 q^{36} + 7993388 q^{61} - 22062592 q^{76} - 19131876 q^{81} - 50334444 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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 \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(3\) 33.0681 − 33.0681i 0.707107 − 0.707107i
\(4\) 128.000i 1.00000i
\(5\) 0 0
\(6\) 0 0
\(7\) 927.132 + 927.132i 1.02164 + 1.02164i 0.999761 + 0.0218805i \(0.00696535\pi\)
0.0218805 + 0.999761i \(0.493035\pi\)
\(8\) 0 0
\(9\) − 2187.00i − 1.00000i
\(10\) 0 0
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) 4232.72 + 4232.72i 0.707107 + 0.707107i
\(13\) −6786.31 + 6786.31i −0.856706 + 0.856706i −0.990949 − 0.134242i \(-0.957140\pi\)
0.134242 + 0.990949i \(0.457140\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −16384.0 −1.00000
\(17\) 0 0 −0.707107 − 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(18\) 0 0
\(19\) 43091.0i 1.44128i 0.693308 + 0.720641i \(0.256152\pi\)
−0.693308 + 0.720641i \(0.743848\pi\)
\(20\) 0 0
\(21\) 61317.0 1.44482
\(22\) 0 0
\(23\) 0 0 0.707107 − 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) −72320.0 − 72320.0i −0.707107 − 0.707107i
\(28\) −118673. + 118673.i −1.02164 + 1.02164i
\(29\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) 0 0
\(31\) 331387. 1.99788 0.998940 − 0.0460243i \(-0.0146552\pi\)
0.998940 + 0.0460243i \(0.0146552\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 279936. 1.00000
\(37\) 388259. + 388259.i 1.26013 + 1.26013i 0.951029 + 0.309100i \(0.100028\pi\)
0.309100 + 0.951029i \(0.399972\pi\)
\(38\) 0 0
\(39\) 448821.i 1.21157i
\(40\) 0 0
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) 0 0
\(43\) −678081. + 678081.i −1.30060 + 1.30060i −0.372605 + 0.927990i \(0.621535\pi\)
−0.927990 + 0.372605i \(0.878465\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 −0.707107 − 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(48\) −541788. + 541788.i −0.707107 + 0.707107i
\(49\) 895604.i 1.08750i
\(50\) 0 0
\(51\) 0 0
\(52\) −868648. − 868648.i −0.856706 − 0.856706i
\(53\) 0 0 0.707107 − 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 1.42494e6 + 1.42494e6i 1.01914 + 1.01914i
\(58\) 0 0
\(59\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) 1.99835e6 1.12724 0.563620 − 0.826034i \(-0.309408\pi\)
0.563620 + 0.826034i \(0.309408\pi\)
\(62\) 0 0
\(63\) 2.02764e6 − 2.02764e6i 1.02164 − 1.02164i
\(64\) − 2.09715e6i − 1.00000i
\(65\) 0 0
\(66\) 0 0
\(67\) −1.97127e6 − 1.97127e6i −0.800726 − 0.800726i 0.182483 − 0.983209i \(-0.441586\pi\)
−0.983209 + 0.182483i \(0.941586\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 0 0
\(73\) 1.55183e6 − 1.55183e6i 0.466890 − 0.466890i −0.434016 − 0.900905i \(-0.642904\pi\)
0.900905 + 0.434016i \(0.142904\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) −5.51565e6 −1.44128
\(77\) 0 0
\(78\) 0 0
\(79\) − 8.76304e6i − 1.99968i −0.0179303 − 0.999839i \(-0.505708\pi\)
0.0179303 − 0.999839i \(-0.494292\pi\)
\(80\) 0 0
\(81\) −4.78297e6 −1.00000
\(82\) 0 0
\(83\) 0 0 0.707107 − 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(84\) 7.84858e6i 1.44482i
\(85\) 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
\(91\) −1.25836e7 −1.75049
\(92\) 0 0
\(93\) 1.09583e7 − 1.09583e7i 1.41271 − 1.41271i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 2.84502e6 + 2.84502e6i 0.316507 + 0.316507i 0.847424 − 0.530917i \(-0.178152\pi\)
−0.530917 + 0.847424i \(0.678152\pi\)
\(98\) 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 75.8.e.b.32.2 yes 4
3.2 odd 2 CM 75.8.e.b.32.2 yes 4
5.2 odd 4 inner 75.8.e.b.68.1 yes 4
5.3 odd 4 inner 75.8.e.b.68.2 yes 4
5.4 even 2 inner 75.8.e.b.32.1 ✓ 4
15.2 even 4 inner 75.8.e.b.68.1 yes 4
15.8 even 4 inner 75.8.e.b.68.2 yes 4
15.14 odd 2 inner 75.8.e.b.32.1 ✓ 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
75.8.e.b.32.1 ✓ 4 5.4 even 2 inner
75.8.e.b.32.1 ✓ 4 15.14 odd 2 inner
75.8.e.b.32.2 yes 4 1.1 even 1 trivial
75.8.e.b.32.2 yes 4 3.2 odd 2 CM
75.8.e.b.68.1 yes 4 5.2 odd 4 inner
75.8.e.b.68.1 yes 4 15.2 even 4 inner
75.8.e.b.68.2 yes 4 5.3 odd 4 inner
75.8.e.b.68.2 yes 4 15.8 even 4 inner