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Properties

Label 704.2.e.c.703.3
Level $704$
Weight $2$
Character 704.703
Analytic conductor $5.621$
Analytic rank $0$
Dimension $4$
CM discriminant -11
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] = [704,2,Mod(703,704)] 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("704.703"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(704, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 704 = 2^{6} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 704.e (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,0,6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(5.62146830230\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{-11})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 2x^{2} - 3x + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 176)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 703.3
Root \(1.68614 - 0.396143i\) of defining polynomial
Character \(\chi\) \(=\) 704.703
Dual form 704.2.e.c.703.2

$q$-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)
 
\(f(q)\) \(=\) \(q+0.792287i q^{3} -1.37228 q^{5} +2.37228 q^{9} +3.31662i q^{11} -1.08724i q^{15} +6.13592i q^{23} -3.11684 q^{25} +4.25639i q^{27} +9.30506i q^{31} -2.62772 q^{33} +12.1168 q^{37} -3.25544 q^{45} +6.63325i q^{47} -7.00000 q^{49} -6.00000 q^{53} -4.55134i q^{55} +14.6487i q^{59} -16.2333i q^{67} -4.86141 q^{69} -10.8896i q^{71} -2.46943i q^{75} +3.74456 q^{81} +18.8614 q^{89} -7.37228 q^{93} -0.116844 q^{97} +7.86797i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 6 q^{5} - 2 q^{9} + 22 q^{25} - 22 q^{33} + 14 q^{37} - 36 q^{45} - 28 q^{49} - 24 q^{53} + 38 q^{69} - 8 q^{81} + 18 q^{89} - 18 q^{93} + 34 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/704\mathbb{Z}\right)^\times\).

\(n\) \(133\) \(321\) \(639\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

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
\(3\) 0.792287i 0.457427i 0.973494 + 0.228714i \(0.0734519\pi\)
−0.973494 + 0.228714i \(0.926548\pi\)
\(4\) 0 0
\(5\) −1.37228 −0.613703 −0.306851 − 0.951757i \(-0.599275\pi\)
−0.306851 + 0.951757i \(0.599275\pi\)
\(6\) 0 0
\(7\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(8\) 0 0
\(9\) 2.37228 0.790760
\(10\) 0 0
\(11\) 3.31662i 1.00000i
\(12\) 0 0
\(13\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(14\) 0 0
\(15\) − 1.08724i − 0.280724i
\(16\) 0 0
\(17\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(18\) 0 0
\(19\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 6.13592i 1.27943i 0.768613 + 0.639713i \(0.220947\pi\)
−0.768613 + 0.639713i \(0.779053\pi\)
\(24\) 0 0
\(25\) −3.11684 −0.623369
\(26\) 0 0
\(27\) 4.25639i 0.819142i
\(28\) 0 0
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) 9.30506i 1.67124i 0.549309 + 0.835619i \(0.314891\pi\)
−0.549309 + 0.835619i \(0.685109\pi\)
\(32\) 0 0
\(33\) −2.62772 −0.457427
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 12.1168 1.99200 0.995998 − 0.0893706i \(-0.0284856\pi\)
0.995998 + 0.0893706i \(0.0284856\pi\)
\(38\) 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
\(45\) −3.25544 −0.485292
\(46\) 0 0
\(47\) 6.63325i 0.967559i 0.875190 + 0.483779i \(0.160736\pi\)
−0.875190 + 0.483779i \(0.839264\pi\)
\(48\) 0 0
\(49\) −7.00000 −1.00000
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −6.00000 −0.824163 −0.412082 − 0.911147i \(-0.635198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) 0 0
\(55\) − 4.55134i − 0.613703i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 14.6487i 1.90710i 0.301239 + 0.953549i \(0.402600\pi\)
−0.301239 + 0.953549i \(0.597400\pi\)
\(60\) 0 0
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) − 16.2333i − 1.98321i −0.129307 − 0.991605i \(-0.541275\pi\)
0.129307 − 0.991605i \(-0.458725\pi\)
\(68\) 0 0
\(69\) −4.86141 −0.585245
\(70\) 0 0
\(71\) − 10.8896i − 1.29236i −0.763184 − 0.646181i \(-0.776365\pi\)
0.763184 − 0.646181i \(-0.223635\pi\)
\(72\) 0 0
\(73\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(74\) 0 0
\(75\) − 2.46943i − 0.285146i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) 0 0
\(81\) 3.74456 0.416063
\(82\) 0 0
\(83\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 18.8614 1.99931 0.999653 − 0.0263586i \(-0.00839118\pi\)
0.999653 + 0.0263586i \(0.00839118\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) −7.37228 −0.764470
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −0.116844 −0.0118637 −0.00593185 − 0.999982i \(-0.501888\pi\)
−0.00593185 + 0.999982i \(0.501888\pi\)
\(98\) 0 0
\(99\) 7.86797i 0.790760i
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 704.2.e.c.703.3 4
4.3 odd 2 inner 704.2.e.c.703.2 4
8.3 odd 2 176.2.e.b.175.3 yes 4
8.5 even 2 176.2.e.b.175.2 ✓ 4
11.10 odd 2 CM 704.2.e.c.703.3 4
16.3 odd 4 2816.2.g.c.1407.6 8
16.5 even 4 2816.2.g.c.1407.5 8
16.11 odd 4 2816.2.g.c.1407.3 8
16.13 even 4 2816.2.g.c.1407.4 8
24.5 odd 2 1584.2.o.e.703.2 4
24.11 even 2 1584.2.o.e.703.1 4
44.43 even 2 inner 704.2.e.c.703.2 4
88.21 odd 2 176.2.e.b.175.2 ✓ 4
88.43 even 2 176.2.e.b.175.3 yes 4
176.21 odd 4 2816.2.g.c.1407.5 8
176.43 even 4 2816.2.g.c.1407.3 8
176.109 odd 4 2816.2.g.c.1407.4 8
176.131 even 4 2816.2.g.c.1407.6 8
264.131 odd 2 1584.2.o.e.703.1 4
264.197 even 2 1584.2.o.e.703.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
176.2.e.b.175.2 ✓ 4 8.5 even 2
176.2.e.b.175.2 ✓ 4 88.21 odd 2
176.2.e.b.175.3 yes 4 8.3 odd 2
176.2.e.b.175.3 yes 4 88.43 even 2
704.2.e.c.703.2 4 4.3 odd 2 inner
704.2.e.c.703.2 4 44.43 even 2 inner
704.2.e.c.703.3 4 1.1 even 1 trivial
704.2.e.c.703.3 4 11.10 odd 2 CM
1584.2.o.e.703.1 4 24.11 even 2
1584.2.o.e.703.1 4 264.131 odd 2
1584.2.o.e.703.2 4 24.5 odd 2
1584.2.o.e.703.2 4 264.197 even 2
2816.2.g.c.1407.3 8 16.11 odd 4
2816.2.g.c.1407.3 8 176.43 even 4
2816.2.g.c.1407.4 8 16.13 even 4
2816.2.g.c.1407.4 8 176.109 odd 4
2816.2.g.c.1407.5 8 16.5 even 4
2816.2.g.c.1407.5 8 176.21 odd 4
2816.2.g.c.1407.6 8 16.3 odd 4
2816.2.g.c.1407.6 8 176.131 even 4