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Properties

Label 49.4.c.d.18.1
Level $49$
Weight $4$
Character 49.18
Analytic conductor $2.891$
Analytic rank $0$
Dimension $2$
CM discriminant -7
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] = [49,4,Mod(18,49)] 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("49.18"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(49, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([4])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 49 = 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 49.c (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,5,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(2.89109359028\)
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: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{3}]$

Embedding invariants

Embedding label 18.1
Root \(0.500000 + 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 49.18
Dual form 49.4.c.d.30.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(2.50000 + 4.33013i) q^{2} +(-8.50000 + 14.7224i) q^{4} -45.0000 q^{8} +(13.5000 + 23.3827i) q^{9} +(34.0000 - 58.8897i) q^{11} +(-44.5000 - 77.0763i) q^{16} +(-67.5000 + 116.913i) q^{18} +340.000 q^{22} +(20.0000 + 34.6410i) q^{23} +(62.5000 - 108.253i) q^{25} -166.000 q^{29} +(42.5000 - 73.6122i) q^{32} -459.000 q^{36} +(-225.000 - 389.711i) q^{37} -180.000 q^{43} +(578.000 + 1001.13i) q^{44} +(-100.000 + 173.205i) q^{46} +625.000 q^{50} +(-295.000 + 510.955i) q^{53} +(-415.000 - 718.801i) q^{58} -287.000 q^{64} +(370.000 - 640.859i) q^{67} +688.000 q^{71} +(-607.500 - 1052.22i) q^{72} +(1125.00 - 1948.56i) q^{74} +(692.000 + 1198.58i) q^{79} +(-364.500 + 631.333i) q^{81} +(-450.000 - 779.423i) q^{86} +(-1530.00 + 2650.04i) q^{88} -680.000 q^{92} +1836.00 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 5 q^{2} - 17 q^{4} - 90 q^{8} + 27 q^{9} + 68 q^{11} - 89 q^{16} - 135 q^{18} + 680 q^{22} + 40 q^{23} + 125 q^{25} - 332 q^{29} + 85 q^{32} - 918 q^{36} - 450 q^{37} - 360 q^{43} + 1156 q^{44} - 200 q^{46}+ \cdots + 3672 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/49\mathbb{Z}\right)^\times\).

\(n\) \(3\)
\(\chi(n)\) \(e\left(\frac{2}{3}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.50000 + 4.33013i 0.883883 + 1.53093i 0.846988 + 0.531612i \(0.178414\pi\)
0.0368954 + 0.999319i \(0.488253\pi\)
\(3\) 0 0 −0.866025 − 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(4\) −8.50000 + 14.7224i −1.06250 + 1.84030i
\(5\) 0 0 0.866025 − 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) −45.0000 −1.98874
\(9\) 13.5000 + 23.3827i 0.500000 + 0.866025i
\(10\) 0 0
\(11\) 34.0000 − 58.8897i 0.931944 − 1.61417i 0.151950 − 0.988388i \(-0.451445\pi\)
0.779994 − 0.625786i \(-0.215222\pi\)
\(12\) 0 0
\(13\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −44.5000 − 77.0763i −0.695312 − 1.20432i
\(17\) 0 0 −0.866025 − 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(18\) −67.5000 + 116.913i −0.883883 + 1.53093i
\(19\) 0 0 0.866025 − 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 340.000 3.29492
\(23\) 20.0000 + 34.6410i 0.181317 + 0.314050i 0.942329 − 0.334687i \(-0.108631\pi\)
−0.761012 + 0.648737i \(0.775297\pi\)
\(24\) 0 0
\(25\) 62.5000 − 108.253i 0.500000 − 0.866025i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −166.000 −1.06295 −0.531473 − 0.847075i \(-0.678361\pi\)
−0.531473 + 0.847075i \(0.678361\pi\)
\(30\) 0 0
\(31\) 0 0 −0.866025 − 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(32\) 42.5000 − 73.6122i 0.234782 − 0.406654i
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) −459.000 −2.12500
\(37\) −225.000 − 389.711i −0.999724 − 1.73157i −0.520223 − 0.854030i \(-0.674151\pi\)
−0.479500 − 0.877542i \(-0.659182\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) 0 0
\(43\) −180.000 −0.638366 −0.319183 − 0.947693i \(-0.603408\pi\)
−0.319183 + 0.947693i \(0.603408\pi\)
\(44\) 578.000 + 1001.13i 1.98038 + 3.43012i
\(45\) 0 0
\(46\) −100.000 + 173.205i −0.320526 + 0.555167i
\(47\) 0 0 0.866025 − 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 625.000 1.76777
\(51\) 0 0
\(52\) 0 0
\(53\) −295.000 + 510.955i −0.764554 + 1.32425i 0.175928 + 0.984403i \(0.443707\pi\)
−0.940482 + 0.339843i \(0.889626\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) −415.000 − 718.801i −0.939520 − 1.62730i
\(59\) 0 0 −0.866025 − 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(60\) 0 0
\(61\) 0 0 0.866025 − 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) −287.000 −0.560547
\(65\) 0 0
\(66\) 0 0
\(67\) 370.000 − 640.859i 0.674667 − 1.16856i −0.301899 − 0.953340i \(-0.597621\pi\)
0.976566 − 0.215218i \(-0.0690461\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 688.000 1.15001 0.575004 − 0.818151i \(-0.305000\pi\)
0.575004 + 0.818151i \(0.305000\pi\)
\(72\) −607.500 − 1052.22i −0.994369 − 1.72230i
\(73\) 0 0 −0.866025 − 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(74\) 1125.00 − 1948.56i 1.76728 − 3.06102i
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 692.000 + 1198.58i 0.985520 + 1.70697i 0.639602 + 0.768706i \(0.279099\pi\)
0.345918 + 0.938265i \(0.387568\pi\)
\(80\) 0 0
\(81\) −364.500 + 631.333i −0.500000 + 0.866025i
\(82\) 0 0
\(83\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −450.000 − 779.423i −0.564241 − 0.977295i
\(87\) 0 0
\(88\) −1530.00 + 2650.04i −1.85339 + 3.21017i
\(89\) 0 0 0.866025 − 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −680.000 −0.770597
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(98\) 0 0
\(99\) 1836.00 1.86389
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 49.4.c.d.18.1 2
3.2 odd 2 441.4.e.a.361.1 2
7.2 even 3 inner 49.4.c.d.30.1 2
7.3 odd 6 49.4.a.a.1.1 ✓ 1
7.4 even 3 49.4.a.a.1.1 ✓ 1
7.5 odd 6 inner 49.4.c.d.30.1 2
7.6 odd 2 CM 49.4.c.d.18.1 2
21.2 odd 6 441.4.e.a.226.1 2
21.5 even 6 441.4.e.a.226.1 2
21.11 odd 6 441.4.a.m.1.1 1
21.17 even 6 441.4.a.m.1.1 1
21.20 even 2 441.4.e.a.361.1 2
28.3 even 6 784.4.a.k.1.1 1
28.11 odd 6 784.4.a.k.1.1 1
35.4 even 6 1225.4.a.l.1.1 1
35.24 odd 6 1225.4.a.l.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
49.4.a.a.1.1 ✓ 1 7.3 odd 6
49.4.a.a.1.1 ✓ 1 7.4 even 3
49.4.c.d.18.1 2 1.1 even 1 trivial
49.4.c.d.18.1 2 7.6 odd 2 CM
49.4.c.d.30.1 2 7.2 even 3 inner
49.4.c.d.30.1 2 7.5 odd 6 inner
441.4.a.m.1.1 1 21.11 odd 6
441.4.a.m.1.1 1 21.17 even 6
441.4.e.a.226.1 2 21.2 odd 6
441.4.e.a.226.1 2 21.5 even 6
441.4.e.a.361.1 2 3.2 odd 2
441.4.e.a.361.1 2 21.20 even 2
784.4.a.k.1.1 1 28.3 even 6
784.4.a.k.1.1 1 28.11 odd 6
1225.4.a.l.1.1 1 35.4 even 6
1225.4.a.l.1.1 1 35.24 odd 6