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Properties

Label 3328.1.h.a.1663.2
Level $3328$
Weight $1$
Character 3328.1663
Analytic conductor $1.661$
Analytic rank $0$
Dimension $2$
Projective image $D_{2}$
CM/RM discs -4, -52, 13
Inner twists $8$

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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] = [3328,1,Mod(1663,3328)] 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("3328.1663"); S:= CuspForms(chi, 1); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3328, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1, 1])) B = ModularForms(chi, 1).cuspidal_submodule().basis() N = [B[i] for i in range(len(B))]
 
Level: \( N \) \(=\) \( 3328 = 2^{8} \cdot 13 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3328.h (of order \(2\), degree \(1\), not 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: \(1.66088836204\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 208)
Projective image: \(D_{2}\)
Projective field: Galois closure of \(\Q(i, \sqrt{13})\)
Artin image: $D_4:C_2$
Artin field: Galois closure of 8.0.2835349504.2

Embedding invariants

Embedding label 1663.2
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 3328.1663
Dual form 3328.1.h.a.1663.1

$q$-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)
 
\(f(q)\) \(=\) \(q-1.00000 q^{9} +1.00000i q^{13} -2.00000 q^{17} -1.00000 q^{25} +2.00000i q^{29} -1.00000 q^{49} +2.00000i q^{53} -2.00000i q^{61} +1.00000 q^{81} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{9} - 4 q^{17} - 2 q^{25} - 2 q^{49} + 2 q^{81}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3328\mathbb{Z}\right)^\times\).

\(n\) \(261\) \(769\) \(1535\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\)

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
\(3\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(4\) 0 0
\(5\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(6\) 0 0
\(7\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(8\) 0 0
\(9\) −1.00000 −1.00000
\(10\) 0 0
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) 0 0
\(13\) 1.00000i 1.00000i
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −2.00000 −2.00000 −1.00000 \(\pi\)
−1.00000 \(\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(24\) 0 0
\(25\) −1.00000 −1.00000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 2.00000i 2.00000i 1.00000i \(0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) 0 0
\(31\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) 0 0
\(43\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 0 0
\(49\) −1.00000 −1.00000
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 2.00000i 2.00000i 1.00000i \(0.5\pi\)
1.00000i \(0.5\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(60\) 0 0
\(61\) − 2.00000i − 2.00000i − 1.00000i \(-0.5\pi\)
− 1.00000i \(-0.5\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(80\) 0 0
\(81\) 1.00000 1.00000
\(82\) 0 0
\(83\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(98\) 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 3328.1.h.a.1663.2 2
4.3 odd 2 CM 3328.1.h.a.1663.2 2
8.3 odd 2 inner 3328.1.h.a.1663.1 2
8.5 even 2 inner 3328.1.h.a.1663.1 2
13.12 even 2 RM 3328.1.h.a.1663.2 2
16.3 odd 4 208.1.c.a.207.1 ✓ 1
16.5 even 4 832.1.c.a.831.1 1
16.11 odd 4 832.1.c.a.831.1 1
16.13 even 4 208.1.c.a.207.1 ✓ 1
48.29 odd 4 1872.1.i.b.415.1 1
48.35 even 4 1872.1.i.b.415.1 1
52.51 odd 2 CM 3328.1.h.a.1663.2 2
104.51 odd 2 inner 3328.1.h.a.1663.1 2
104.77 even 2 inner 3328.1.h.a.1663.1 2
208.3 odd 12 2704.1.y.b.2175.1 2
208.19 even 12 2704.1.bb.a.991.1 2
208.29 even 12 2704.1.y.b.2175.1 2
208.35 odd 12 2704.1.y.b.1375.1 2
208.45 odd 12 2704.1.bb.a.991.1 2
208.51 odd 4 208.1.c.a.207.1 ✓ 1
208.61 even 12 2704.1.y.b.1375.1 2
208.67 even 12 2704.1.bb.a.191.1 2
208.77 even 4 208.1.c.a.207.1 ✓ 1
208.83 even 4 2704.1.d.a.2367.1 1
208.93 odd 12 2704.1.bb.a.191.1 2
208.99 even 4 2704.1.d.a.2367.1 1
208.109 odd 4 2704.1.d.a.2367.1 1
208.115 even 12 2704.1.bb.a.191.1 2
208.125 odd 4 2704.1.d.a.2367.1 1
208.141 odd 12 2704.1.bb.a.191.1 2
208.147 odd 12 2704.1.y.b.1375.1 2
208.155 odd 4 832.1.c.a.831.1 1
208.163 even 12 2704.1.bb.a.991.1 2
208.173 even 12 2704.1.y.b.1375.1 2
208.179 odd 12 2704.1.y.b.2175.1 2
208.181 even 4 832.1.c.a.831.1 1
208.189 odd 12 2704.1.bb.a.991.1 2
208.205 even 12 2704.1.y.b.2175.1 2
624.77 odd 4 1872.1.i.b.415.1 1
624.467 even 4 1872.1.i.b.415.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
208.1.c.a.207.1 ✓ 1 16.3 odd 4
208.1.c.a.207.1 ✓ 1 16.13 even 4
208.1.c.a.207.1 ✓ 1 208.51 odd 4
208.1.c.a.207.1 ✓ 1 208.77 even 4
832.1.c.a.831.1 1 16.5 even 4
832.1.c.a.831.1 1 16.11 odd 4
832.1.c.a.831.1 1 208.155 odd 4
832.1.c.a.831.1 1 208.181 even 4
1872.1.i.b.415.1 1 48.29 odd 4
1872.1.i.b.415.1 1 48.35 even 4
1872.1.i.b.415.1 1 624.77 odd 4
1872.1.i.b.415.1 1 624.467 even 4
2704.1.d.a.2367.1 1 208.83 even 4
2704.1.d.a.2367.1 1 208.99 even 4
2704.1.d.a.2367.1 1 208.109 odd 4
2704.1.d.a.2367.1 1 208.125 odd 4
2704.1.y.b.1375.1 2 208.35 odd 12
2704.1.y.b.1375.1 2 208.61 even 12
2704.1.y.b.1375.1 2 208.147 odd 12
2704.1.y.b.1375.1 2 208.173 even 12
2704.1.y.b.2175.1 2 208.3 odd 12
2704.1.y.b.2175.1 2 208.29 even 12
2704.1.y.b.2175.1 2 208.179 odd 12
2704.1.y.b.2175.1 2 208.205 even 12
2704.1.bb.a.191.1 2 208.67 even 12
2704.1.bb.a.191.1 2 208.93 odd 12
2704.1.bb.a.191.1 2 208.115 even 12
2704.1.bb.a.191.1 2 208.141 odd 12
2704.1.bb.a.991.1 2 208.19 even 12
2704.1.bb.a.991.1 2 208.45 odd 12
2704.1.bb.a.991.1 2 208.163 even 12
2704.1.bb.a.991.1 2 208.189 odd 12
3328.1.h.a.1663.1 2 8.3 odd 2 inner
3328.1.h.a.1663.1 2 8.5 even 2 inner
3328.1.h.a.1663.1 2 104.51 odd 2 inner
3328.1.h.a.1663.1 2 104.77 even 2 inner
3328.1.h.a.1663.2 2 1.1 even 1 trivial
3328.1.h.a.1663.2 2 4.3 odd 2 CM
3328.1.h.a.1663.2 2 13.12 even 2 RM
3328.1.h.a.1663.2 2 52.51 odd 2 CM