Properties

Label 1521.1.bm.a.1513.1
Level $1521$
Weight $1$
Character 1521.1513
Analytic conductor $0.759$
Analytic rank $0$
Dimension $24$
Projective image $D_{52}$
CM discriminant -3
Inner twists $4$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1521,1,Mod(73,1521)] 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("1521.73"); S:= CuspForms(chi, 1); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1521, base_ring=CyclotomicField(52)) chi = DirichletCharacter(H, H._module([0, 17])) B = ModularForms(chi, 1).cuspidal_submodule().basis() N = [B[i] for i in range(len(B))]
 
Level: \( N \) \(=\) \( 1521 = 3^{2} \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1521.bm (of order \(52\), degree \(24\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(0.759077884215\)
Analytic rank: \(0\)
Dimension: \(24\)
Coefficient field: \(\Q(\zeta_{52})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{24} - x^{22} + x^{20} - x^{18} + x^{16} - x^{14} + x^{12} - x^{10} + x^{8} - x^{6} + x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{52}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{52} - \cdots)\)

Embedding invariants

Embedding label 1513.1
Root \(0.935016 - 0.354605i\) of defining polynomial
Character \(\chi\) \(=\) 1521.1513
Dual form 1521.1.bm.a.190.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.992709 + 0.120537i) q^{4} +(-0.328749 + 1.79393i) q^{7} -1.00000i q^{13} +(0.970942 + 0.239316i) q^{16} +(0.0853881 - 0.0853881i) q^{19} +(0.464723 + 0.885456i) q^{25} +(-0.542586 + 1.74122i) q^{28} +(-1.17759 + 0.366951i) q^{31} +(-0.107253 - 0.344186i) q^{37} +(0.764919 + 1.45743i) q^{43} +(-2.17508 - 0.824898i) q^{49} +(0.120537 - 0.992709i) q^{52} +(1.12785 - 1.63397i) q^{61} +(0.935016 + 0.354605i) q^{64} +(-0.468379 - 0.366951i) q^{67} +(-0.0624722 - 1.03279i) q^{73} +(0.0950579 - 0.0744731i) q^{76} +(-0.225408 - 1.85640i) q^{79} +(1.79393 + 0.328749i) q^{91} +(-1.21026 + 0.731626i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 24 q + 2 q^{7} + 2 q^{16} + 2 q^{19} - 2 q^{28} - 2 q^{31} - 2 q^{37} - 2 q^{52} + 2 q^{67} + 2 q^{73} + 2 q^{76} + 2 q^{91} - 2 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1521\mathbb{Z}\right)^\times\).

\(n\) \(677\) \(847\)
\(\chi(n)\) \(1\) \(e\left(\frac{27}{52}\right)\)

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 −0.998176 − 0.0603785i \(-0.980769\pi\)
0.998176 + 0.0603785i \(0.0192308\pi\)
\(3\) 0 0
\(4\) 0.992709 + 0.120537i 0.992709 + 0.120537i
\(5\) 0 0 −0.855781 − 0.517338i \(-0.826923\pi\)
0.855781 + 0.517338i \(0.173077\pi\)
\(6\) 0 0
\(7\) −0.328749 + 1.79393i −0.328749 + 1.79393i 0.239316 + 0.970942i \(0.423077\pi\)
−0.568065 + 0.822984i \(0.692308\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 0 0 0.998176 − 0.0603785i \(-0.0192308\pi\)
−0.998176 + 0.0603785i \(0.980769\pi\)
\(12\) 0 0
\(13\) − 1.00000i − 1.00000i
\(14\) 0 0
\(15\) 0 0
\(16\) 0.970942 + 0.239316i 0.970942 + 0.239316i
\(17\) 0 0 −0.568065 − 0.822984i \(-0.692308\pi\)
0.568065 + 0.822984i \(0.307692\pi\)
\(18\) 0 0
\(19\) 0.0853881 − 0.0853881i 0.0853881 − 0.0853881i −0.663123 − 0.748511i \(-0.730769\pi\)
0.748511 + 0.663123i \(0.230769\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(24\) 0 0
\(25\) 0.464723 + 0.885456i 0.464723 + 0.885456i
\(26\) 0 0
\(27\) 0 0
\(28\) −0.542586 + 1.74122i −0.542586 + 1.74122i
\(29\) 0 0 −0.663123 − 0.748511i \(-0.730769\pi\)
0.663123 + 0.748511i \(0.269231\pi\)
\(30\) 0 0
\(31\) −1.17759 + 0.366951i −1.17759 + 0.366951i −0.822984 − 0.568065i \(-0.807692\pi\)
−0.354605 + 0.935016i \(0.615385\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −0.107253 − 0.344186i −0.107253 − 0.344186i 0.885456 − 0.464723i \(-0.153846\pi\)
−0.992709 + 0.120537i \(0.961538\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 0.410413 − 0.911900i \(-0.365385\pi\)
−0.410413 + 0.911900i \(0.634615\pi\)
\(42\) 0 0
\(43\) 0.764919 + 1.45743i 0.764919 + 1.45743i 0.885456 + 0.464723i \(0.153846\pi\)
−0.120537 + 0.992709i \(0.538462\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 −0.787183 − 0.616719i \(-0.788462\pi\)
0.787183 + 0.616719i \(0.211538\pi\)
\(48\) 0 0
\(49\) −2.17508 − 0.824898i −2.17508 − 0.824898i
\(50\) 0 0
\(51\) 0 0
\(52\) 0.120537 − 0.992709i 0.120537 − 0.992709i
\(53\) 0 0 0.822984 − 0.568065i \(-0.192308\pi\)
−0.822984 + 0.568065i \(0.807692\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 0.517338 − 0.855781i \(-0.326923\pi\)
−0.517338 + 0.855781i \(0.673077\pi\)
\(60\) 0 0
\(61\) 1.12785 − 1.63397i 1.12785 − 1.63397i 0.464723 − 0.885456i \(-0.346154\pi\)
0.663123 − 0.748511i \(-0.269231\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0.935016 + 0.354605i 0.935016 + 0.354605i
\(65\) 0 0
\(66\) 0 0
\(67\) −0.468379 − 0.366951i −0.468379 − 0.366951i 0.354605 − 0.935016i \(-0.384615\pi\)
−0.822984 + 0.568065i \(0.807692\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 0.410413 − 0.911900i \(-0.365385\pi\)
−0.410413 + 0.911900i \(0.634615\pi\)
\(72\) 0 0
\(73\) −0.0624722 − 1.03279i −0.0624722 − 1.03279i −0.885456 − 0.464723i \(-0.846154\pi\)
0.822984 − 0.568065i \(-0.192308\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0.0950579 − 0.0744731i 0.0950579 − 0.0744731i
\(77\) 0 0
\(78\) 0 0
\(79\) −0.225408 − 1.85640i −0.225408 − 1.85640i −0.464723 − 0.885456i \(-0.653846\pi\)
0.239316 − 0.970942i \(-0.423077\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 0 0 0.911900 − 0.410413i \(-0.134615\pi\)
−0.911900 + 0.410413i \(0.865385\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 0.707107 − 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(90\) 0 0
\(91\) 1.79393 + 0.328749i 1.79393 + 0.328749i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −1.21026 + 0.731626i −1.21026 + 0.731626i −0.970942 − 0.239316i \(-0.923077\pi\)
−0.239316 + 0.970942i \(0.576923\pi\)
\(98\) 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 1521.1.bm.a.1513.1 yes 24
3.2 odd 2 CM 1521.1.bm.a.1513.1 yes 24
169.21 odd 52 inner 1521.1.bm.a.190.1 ✓ 24
507.359 even 52 inner 1521.1.bm.a.190.1 ✓ 24
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1521.1.bm.a.190.1 ✓ 24 169.21 odd 52 inner
1521.1.bm.a.190.1 ✓ 24 507.359 even 52 inner
1521.1.bm.a.1513.1 yes 24 1.1 even 1 trivial
1521.1.bm.a.1513.1 yes 24 3.2 odd 2 CM