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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1575,2,Mod(26,1575)] 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("1575.26"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1575, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 0, 5])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1575 = 3^{2} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1575.bk (of order \(6\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,-4,0,0,8,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(12.5764383184\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{-3})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{31}]\)
Coefficient ring index: \( 3^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 1151.1
Root \(-1.22474 - 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 1575.1151
Dual form 1575.2.bk.b.26.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-1.00000 + 1.73205i) q^{4} +(2.00000 + 1.73205i) q^{7} +(-3.67423 - 2.12132i) q^{11} -3.46410i q^{13} +(-2.00000 - 3.46410i) q^{16} +(3.67423 - 6.36396i) q^{17} +(6.00000 - 3.46410i) q^{19} +(-7.34847 + 4.24264i) q^{23} +(-5.00000 + 1.73205i) q^{28} -8.48528i q^{29} +(4.50000 + 2.59808i) q^{31} +(-0.500000 - 0.866025i) q^{37} +7.34847 q^{41} +1.00000 q^{43} +(7.34847 - 4.24264i) q^{44} +(3.67423 + 6.36396i) q^{47} +(1.00000 + 6.92820i) q^{49} +(6.00000 + 3.46410i) q^{52} +(3.67423 + 2.12132i) q^{53} +(3.67423 - 6.36396i) q^{59} +(-1.50000 + 0.866025i) q^{61} +8.00000 q^{64} +(-1.00000 + 1.73205i) q^{67} +(7.34847 + 12.7279i) q^{68} -4.24264i q^{71} +(-7.50000 - 4.33013i) q^{73} +13.8564i q^{76} +(-3.67423 - 10.6066i) q^{77} +(6.50000 + 11.2583i) q^{79} -7.34847 q^{83} +(-3.67423 - 6.36396i) q^{89} +(6.00000 - 6.92820i) q^{91} -16.9706i q^{92} +1.73205i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{4} + 8 q^{7} - 8 q^{16} + 24 q^{19} - 20 q^{28} + 18 q^{31} - 2 q^{37} + 4 q^{43} + 4 q^{49} + 24 q^{52} - 6 q^{61} + 32 q^{64} - 4 q^{67} - 30 q^{73} + 26 q^{79} + 24 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1575\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(451\) \(1226\)
\(\chi(n)\) \(1\) \(e\left(\frac{1}{6}\right)\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0 −0.500000 − 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(3\) 0 0
\(4\) −1.00000 + 1.73205i −0.500000 + 0.866025i
\(5\) 0 0
\(6\) 0 0
\(7\) 2.00000 + 1.73205i 0.755929 + 0.654654i
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −3.67423 − 2.12132i −1.10782 − 0.639602i −0.169559 − 0.985520i \(-0.554234\pi\)
−0.938265 + 0.345918i \(0.887568\pi\)
\(12\) 0 0
\(13\) − 3.46410i − 0.960769i −0.877058 − 0.480384i \(-0.840497\pi\)
0.877058 − 0.480384i \(-0.159503\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −2.00000 − 3.46410i −0.500000 − 0.866025i
\(17\) 3.67423 − 6.36396i 0.891133 − 1.54349i 0.0526138 − 0.998615i \(-0.483245\pi\)
0.838519 − 0.544872i \(-0.183422\pi\)
\(18\) 0 0
\(19\) 6.00000 − 3.46410i 1.37649 − 0.794719i 0.384759 − 0.923017i \(-0.374285\pi\)
0.991736 + 0.128298i \(0.0409513\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −7.34847 + 4.24264i −1.53226 + 0.884652i −0.533005 + 0.846112i \(0.678937\pi\)
−0.999257 + 0.0385394i \(0.987729\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 0 0
\(28\) −5.00000 + 1.73205i −0.944911 + 0.327327i
\(29\) − 8.48528i − 1.57568i −0.615882 − 0.787839i \(-0.711200\pi\)
0.615882 − 0.787839i \(-0.288800\pi\)
\(30\) 0 0
\(31\) 4.50000 + 2.59808i 0.808224 + 0.466628i 0.846339 − 0.532645i \(-0.178802\pi\)
−0.0381148 + 0.999273i \(0.512135\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −0.500000 − 0.866025i −0.0821995 − 0.142374i 0.821995 − 0.569495i \(-0.192861\pi\)
−0.904194 + 0.427121i \(0.859528\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 7.34847 1.14764 0.573819 − 0.818982i \(-0.305461\pi\)
0.573819 + 0.818982i \(0.305461\pi\)
\(42\) 0 0
\(43\) 1.00000 0.152499 0.0762493 − 0.997089i \(-0.475706\pi\)
0.0762493 + 0.997089i \(0.475706\pi\)
\(44\) 7.34847 − 4.24264i 1.10782 − 0.639602i
\(45\) 0 0
\(46\) 0 0
\(47\) 3.67423 + 6.36396i 0.535942 + 0.928279i 0.999117 + 0.0420122i \(0.0133768\pi\)
−0.463175 + 0.886267i \(0.653290\pi\)
\(48\) 0 0
\(49\) 1.00000 + 6.92820i 0.142857 + 0.989743i
\(50\) 0 0
\(51\) 0 0
\(52\) 6.00000 + 3.46410i 0.832050 + 0.480384i
\(53\) 3.67423 + 2.12132i 0.504695 + 0.291386i 0.730650 − 0.682752i \(-0.239217\pi\)
−0.225955 + 0.974138i \(0.572550\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 3.67423 − 6.36396i 0.478345 − 0.828517i −0.521347 − 0.853345i \(-0.674570\pi\)
0.999692 + 0.0248275i \(0.00790366\pi\)
\(60\) 0 0
\(61\) −1.50000 + 0.866025i −0.192055 + 0.110883i −0.592944 − 0.805243i \(-0.702035\pi\)
0.400889 + 0.916127i \(0.368701\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 8.00000 1.00000
\(65\) 0 0
\(66\) 0 0
\(67\) −1.00000 + 1.73205i −0.122169 + 0.211604i −0.920623 − 0.390453i \(-0.872318\pi\)
0.798454 + 0.602056i \(0.205652\pi\)
\(68\) 7.34847 + 12.7279i 0.891133 + 1.54349i
\(69\) 0 0
\(70\) 0 0
\(71\) − 4.24264i − 0.503509i −0.967791 − 0.251754i \(-0.918992\pi\)
0.967791 − 0.251754i \(-0.0810075\pi\)
\(72\) 0 0
\(73\) −7.50000 − 4.33013i −0.877809 − 0.506803i −0.00787336 − 0.999969i \(-0.502506\pi\)
−0.869935 + 0.493166i \(0.835840\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 13.8564i 1.58944i
\(77\) −3.67423 − 10.6066i −0.418718 − 1.20873i
\(78\) 0 0
\(79\) 6.50000 + 11.2583i 0.731307 + 1.26666i 0.956325 + 0.292306i \(0.0944227\pi\)
−0.225018 + 0.974355i \(0.572244\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −7.34847 −0.806599 −0.403300 − 0.915068i \(-0.632137\pi\)
−0.403300 + 0.915068i \(0.632137\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −3.67423 − 6.36396i −0.389468 − 0.674579i 0.602910 − 0.797809i \(-0.294008\pi\)
−0.992378 + 0.123231i \(0.960674\pi\)
\(90\) 0 0
\(91\) 6.00000 − 6.92820i 0.628971 − 0.726273i
\(92\) − 16.9706i − 1.76930i
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 1.73205i 0.175863i 0.996127 + 0.0879316i \(0.0280257\pi\)
−0.996127 + 0.0879316i \(0.971974\pi\)
\(98\) 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 1575.2.bk.b.1151.1 yes 4
3.2 odd 2 inner 1575.2.bk.b.1151.2 yes 4
5.2 odd 4 1575.2.bc.b.899.1 8
5.3 odd 4 1575.2.bc.b.899.3 8
5.4 even 2 1575.2.bk.a.1151.1 yes 4
7.5 odd 6 inner 1575.2.bk.b.26.2 yes 4
15.2 even 4 1575.2.bc.b.899.2 8
15.8 even 4 1575.2.bc.b.899.4 8
15.14 odd 2 1575.2.bk.a.1151.2 yes 4
21.5 even 6 inner 1575.2.bk.b.26.1 yes 4
35.12 even 12 1575.2.bc.b.1349.4 8
35.19 odd 6 1575.2.bk.a.26.2 yes 4
35.33 even 12 1575.2.bc.b.1349.2 8
105.47 odd 12 1575.2.bc.b.1349.3 8
105.68 odd 12 1575.2.bc.b.1349.1 8
105.89 even 6 1575.2.bk.a.26.1 ✓ 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1575.2.bc.b.899.1 8 5.2 odd 4
1575.2.bc.b.899.2 8 15.2 even 4
1575.2.bc.b.899.3 8 5.3 odd 4
1575.2.bc.b.899.4 8 15.8 even 4
1575.2.bc.b.1349.1 8 105.68 odd 12
1575.2.bc.b.1349.2 8 35.33 even 12
1575.2.bc.b.1349.3 8 105.47 odd 12
1575.2.bc.b.1349.4 8 35.12 even 12
1575.2.bk.a.26.1 ✓ 4 105.89 even 6
1575.2.bk.a.26.2 yes 4 35.19 odd 6
1575.2.bk.a.1151.1 yes 4 5.4 even 2
1575.2.bk.a.1151.2 yes 4 15.14 odd 2
1575.2.bk.b.26.1 yes 4 21.5 even 6 inner
1575.2.bk.b.26.2 yes 4 7.5 odd 6 inner
1575.2.bk.b.1151.1 yes 4 1.1 even 1 trivial
1575.2.bk.b.1151.2 yes 4 3.2 odd 2 inner