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Properties

Label 784.3.r.e.79.1
Level $784$
Weight $3$
Character 784.79
Analytic conductor $21.362$
Analytic rank $0$
Dimension $2$
CM discriminant -4
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] = [784,3,Mod(79,784)] 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("784.79"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(784, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 0, 2])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 784 = 2^{4} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 784.r (of order \(6\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,0,6,0,0,0,-9,0,0] 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: \(21.3624527258\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{25}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 16)
Sato-Tate group: $\mathrm{U}(1)[D_{6}]$

Embedding invariants

Embedding label 79.1
Root \(0.500000 + 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 784.79
Dual form 784.3.r.e.655.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(3.00000 - 5.19615i) q^{5} +(-4.50000 + 7.79423i) q^{9} +10.0000 q^{13} +(15.0000 + 25.9808i) q^{17} +(-5.50000 - 9.52628i) q^{25} +42.0000 q^{29} +(35.0000 - 60.6218i) q^{37} +18.0000 q^{41} +(27.0000 + 46.7654i) q^{45} +(-45.0000 - 77.9423i) q^{53} +(11.0000 - 19.0526i) q^{61} +(30.0000 - 51.9615i) q^{65} +(55.0000 + 95.2628i) q^{73} +(-40.5000 - 70.1481i) q^{81} +180.000 q^{85} +(39.0000 - 67.5500i) q^{89} +130.000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 6 q^{5} - 9 q^{9} + 20 q^{13} + 30 q^{17} - 11 q^{25} + 84 q^{29} + 70 q^{37} + 36 q^{41} + 54 q^{45} - 90 q^{53} + 22 q^{61} + 60 q^{65} + 110 q^{73} - 81 q^{81} + 360 q^{85} + 78 q^{89} + 260 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/784\mathbb{Z}\right)^\times\).

\(n\) \(197\) \(687\) \(689\)
\(\chi(n)\) \(1\) \(-1\) \(e\left(\frac{1}{3}\right)\)

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
\(3\) 0 0 −0.500000 − 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(4\) 0 0
\(5\) 3.00000 − 5.19615i 0.600000 − 1.03923i −0.392820 − 0.919615i \(-0.628501\pi\)
0.992820 − 0.119615i \(-0.0381661\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) −4.50000 + 7.79423i −0.500000 + 0.866025i
\(10\) 0 0
\(11\) 0 0 −0.500000 − 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(12\) 0 0
\(13\) 10.0000 0.769231 0.384615 − 0.923077i \(-0.374334\pi\)
0.384615 + 0.923077i \(0.374334\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 15.0000 + 25.9808i 0.882353 + 1.52828i 0.848718 + 0.528846i \(0.177375\pi\)
0.0336351 + 0.999434i \(0.489292\pi\)
\(18\) 0 0
\(19\) 0 0 0.500000 − 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 0.500000 − 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(24\) 0 0
\(25\) −5.50000 − 9.52628i −0.220000 − 0.381051i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 42.0000 1.44828 0.724138 − 0.689655i \(-0.242238\pi\)
0.724138 + 0.689655i \(0.242238\pi\)
\(30\) 0 0
\(31\) 0 0 −0.500000 − 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 35.0000 − 60.6218i 0.945946 − 1.63843i 0.192100 − 0.981375i \(-0.438470\pi\)
0.753846 − 0.657051i \(-0.228196\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 18.0000 0.439024 0.219512 − 0.975610i \(-0.429553\pi\)
0.219512 + 0.975610i \(0.429553\pi\)
\(42\) 0 0
\(43\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(44\) 0 0
\(45\) 27.0000 + 46.7654i 0.600000 + 1.03923i
\(46\) 0 0
\(47\) 0 0 0.500000 − 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −45.0000 − 77.9423i −0.849057 − 1.47061i −0.882051 − 0.471154i \(-0.843838\pi\)
0.0329946 − 0.999456i \(-0.489496\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 −0.500000 − 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(60\) 0 0
\(61\) 11.0000 − 19.0526i 0.180328 − 0.312337i −0.761664 − 0.647972i \(-0.775617\pi\)
0.941992 + 0.335635i \(0.108951\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 30.0000 − 51.9615i 0.461538 − 0.799408i
\(66\) 0 0
\(67\) 0 0 −0.500000 − 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 0 0
\(73\) 55.0000 + 95.2628i 0.753425 + 1.30497i 0.946154 + 0.323718i \(0.104933\pi\)
−0.192729 + 0.981252i \(0.561734\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 0.500000 − 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(80\) 0 0
\(81\) −40.5000 − 70.1481i −0.500000 − 0.866025i
\(82\) 0 0
\(83\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(84\) 0 0
\(85\) 180.000 2.11765
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 39.0000 − 67.5500i 0.438202 − 0.758989i −0.559349 − 0.828932i \(-0.688949\pi\)
0.997551 + 0.0699439i \(0.0222820\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 130.000 1.34021 0.670103 − 0.742268i \(-0.266250\pi\)
0.670103 + 0.742268i \(0.266250\pi\)
\(98\) 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 784.3.r.e.79.1 2
4.3 odd 2 CM 784.3.r.e.79.1 2
7.2 even 3 16.3.c.a.15.1 ✓ 1
7.3 odd 6 784.3.r.d.655.1 2
7.4 even 3 inner 784.3.r.e.655.1 2
7.5 odd 6 784.3.d.b.687.1 1
7.6 odd 2 784.3.r.d.79.1 2
21.2 odd 6 144.3.g.a.127.1 1
28.3 even 6 784.3.r.d.655.1 2
28.11 odd 6 inner 784.3.r.e.655.1 2
28.19 even 6 784.3.d.b.687.1 1
28.23 odd 6 16.3.c.a.15.1 ✓ 1
28.27 even 2 784.3.r.d.79.1 2
35.2 odd 12 400.3.h.a.399.1 2
35.9 even 6 400.3.b.a.351.1 1
35.23 odd 12 400.3.h.a.399.2 2
56.37 even 6 64.3.c.a.63.1 1
56.51 odd 6 64.3.c.a.63.1 1
63.2 odd 6 1296.3.o.b.271.1 2
63.16 even 3 1296.3.o.o.271.1 2
63.23 odd 6 1296.3.o.b.703.1 2
63.58 even 3 1296.3.o.o.703.1 2
84.23 even 6 144.3.g.a.127.1 1
105.2 even 12 3600.3.j.a.1999.1 2
105.23 even 12 3600.3.j.a.1999.2 2
105.44 odd 6 3600.3.e.c.3151.1 1
112.37 even 12 256.3.d.b.127.1 2
112.51 odd 12 256.3.d.b.127.2 2
112.93 even 12 256.3.d.b.127.2 2
112.107 odd 12 256.3.d.b.127.1 2
140.23 even 12 400.3.h.a.399.2 2
140.79 odd 6 400.3.b.a.351.1 1
140.107 even 12 400.3.h.a.399.1 2
168.107 even 6 576.3.g.b.127.1 1
168.149 odd 6 576.3.g.b.127.1 1
252.23 even 6 1296.3.o.b.703.1 2
252.79 odd 6 1296.3.o.o.271.1 2
252.191 even 6 1296.3.o.b.271.1 2
252.247 odd 6 1296.3.o.o.703.1 2
280.37 odd 12 1600.3.h.b.1599.2 2
280.93 odd 12 1600.3.h.b.1599.1 2
280.107 even 12 1600.3.h.b.1599.2 2
280.149 even 6 1600.3.b.b.1151.1 1
280.163 even 12 1600.3.h.b.1599.1 2
280.219 odd 6 1600.3.b.b.1151.1 1
336.107 even 12 2304.3.b.f.127.2 2
336.149 odd 12 2304.3.b.f.127.2 2
336.275 even 12 2304.3.b.f.127.1 2
336.317 odd 12 2304.3.b.f.127.1 2
420.23 odd 12 3600.3.j.a.1999.2 2
420.107 odd 12 3600.3.j.a.1999.1 2
420.359 even 6 3600.3.e.c.3151.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
16.3.c.a.15.1 ✓ 1 7.2 even 3
16.3.c.a.15.1 ✓ 1 28.23 odd 6
64.3.c.a.63.1 1 56.37 even 6
64.3.c.a.63.1 1 56.51 odd 6
144.3.g.a.127.1 1 21.2 odd 6
144.3.g.a.127.1 1 84.23 even 6
256.3.d.b.127.1 2 112.37 even 12
256.3.d.b.127.1 2 112.107 odd 12
256.3.d.b.127.2 2 112.51 odd 12
256.3.d.b.127.2 2 112.93 even 12
400.3.b.a.351.1 1 35.9 even 6
400.3.b.a.351.1 1 140.79 odd 6
400.3.h.a.399.1 2 35.2 odd 12
400.3.h.a.399.1 2 140.107 even 12
400.3.h.a.399.2 2 35.23 odd 12
400.3.h.a.399.2 2 140.23 even 12
576.3.g.b.127.1 1 168.107 even 6
576.3.g.b.127.1 1 168.149 odd 6
784.3.d.b.687.1 1 7.5 odd 6
784.3.d.b.687.1 1 28.19 even 6
784.3.r.d.79.1 2 7.6 odd 2
784.3.r.d.79.1 2 28.27 even 2
784.3.r.d.655.1 2 7.3 odd 6
784.3.r.d.655.1 2 28.3 even 6
784.3.r.e.79.1 2 1.1 even 1 trivial
784.3.r.e.79.1 2 4.3 odd 2 CM
784.3.r.e.655.1 2 7.4 even 3 inner
784.3.r.e.655.1 2 28.11 odd 6 inner
1296.3.o.b.271.1 2 63.2 odd 6
1296.3.o.b.271.1 2 252.191 even 6
1296.3.o.b.703.1 2 63.23 odd 6
1296.3.o.b.703.1 2 252.23 even 6
1296.3.o.o.271.1 2 63.16 even 3
1296.3.o.o.271.1 2 252.79 odd 6
1296.3.o.o.703.1 2 63.58 even 3
1296.3.o.o.703.1 2 252.247 odd 6
1600.3.b.b.1151.1 1 280.149 even 6
1600.3.b.b.1151.1 1 280.219 odd 6
1600.3.h.b.1599.1 2 280.93 odd 12
1600.3.h.b.1599.1 2 280.163 even 12
1600.3.h.b.1599.2 2 280.37 odd 12
1600.3.h.b.1599.2 2 280.107 even 12
2304.3.b.f.127.1 2 336.275 even 12
2304.3.b.f.127.1 2 336.317 odd 12
2304.3.b.f.127.2 2 336.107 even 12
2304.3.b.f.127.2 2 336.149 odd 12
3600.3.e.c.3151.1 1 105.44 odd 6
3600.3.e.c.3151.1 1 420.359 even 6
3600.3.j.a.1999.1 2 105.2 even 12
3600.3.j.a.1999.1 2 420.107 odd 12
3600.3.j.a.1999.2 2 105.23 even 12
3600.3.j.a.1999.2 2 420.23 odd 12