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Properties

Label 49.6.a.b.1.1
Level $49$
Weight $6$
Character 49.1
Self dual yes
Analytic conductor $7.859$
Analytic rank $0$
Dimension $1$
CM discriminant -7
Inner twists $2$

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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] = [49,6,Mod(1,49)] 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("49.1"); S:= CuspForms(chi, 6); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(49, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 6, names="a")
 
Level: \( N \) \(=\) \( 49 = 7^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 49.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,11,0] 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: yes
Analytic conductor: \(7.85880717084\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $N(\mathrm{U}(1))$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 49.1

$q$-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)
 
\(f(q)\) \(=\) \(q+11.0000 q^{2} +89.0000 q^{4} +627.000 q^{8} -243.000 q^{9} -76.0000 q^{11} +4049.00 q^{16} -2673.00 q^{18} -836.000 q^{22} -4952.00 q^{23} -3125.00 q^{25} +7282.00 q^{29} +24475.0 q^{32} -21627.0 q^{36} -8886.00 q^{37} +11748.0 q^{43} -6764.00 q^{44} -54472.0 q^{46} -34375.0 q^{50} +24550.0 q^{53} +80102.0 q^{58} +139657. q^{64} +69364.0 q^{67} -2224.00 q^{71} -152361. q^{72} -97746.0 q^{74} +80168.0 q^{79} +59049.0 q^{81} +129228. q^{86} -47652.0 q^{88} -440728. q^{92} +18468.0 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 11.0000 1.94454 0.972272 − 0.233854i \(-0.0751336\pi\)
0.972272 + 0.233854i \(0.0751336\pi\)
\(3\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(4\) 89.0000 2.78125
\(5\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 627.000 3.46372
\(9\) −243.000 −1.00000
\(10\) 0 0
\(11\) −76.0000 −0.189379 −0.0946895 − 0.995507i \(-0.530186\pi\)
−0.0946895 + 0.995507i \(0.530186\pi\)
\(12\) 0 0
\(13\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 4049.00 3.95410
\(17\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) −2673.00 −1.94454
\(19\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −836.000 −0.368256
\(23\) −4952.00 −1.95192 −0.975958 − 0.217959i \(-0.930060\pi\)
−0.975958 + 0.217959i \(0.930060\pi\)
\(24\) 0 0
\(25\) −3125.00 −1.00000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 7282.00 1.60789 0.803944 − 0.594705i \(-0.202731\pi\)
0.803944 + 0.594705i \(0.202731\pi\)
\(30\) 0 0
\(31\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 24475.0 4.22520
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) −21627.0 −2.78125
\(37\) −8886.00 −1.06709 −0.533546 − 0.845771i \(-0.679141\pi\)
−0.533546 + 0.845771i \(0.679141\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) 0 0
\(43\) 11748.0 0.968931 0.484465 − 0.874810i \(-0.339014\pi\)
0.484465 + 0.874810i \(0.339014\pi\)
\(44\) −6764.00 −0.526710
\(45\) 0 0
\(46\) −54472.0 −3.79559
\(47\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) −34375.0 −1.94454
\(51\) 0 0
\(52\) 0 0
\(53\) 24550.0 1.20050 0.600250 − 0.799813i \(-0.295068\pi\)
0.600250 + 0.799813i \(0.295068\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 80102.0 3.12661
\(59\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 139657. 4.26199
\(65\) 0 0
\(66\) 0 0
\(67\) 69364.0 1.88776 0.943881 − 0.330286i \(-0.107145\pi\)
0.943881 + 0.330286i \(0.107145\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −2224.00 −0.0523587 −0.0261794 − 0.999657i \(-0.508334\pi\)
−0.0261794 + 0.999657i \(0.508334\pi\)
\(72\) −152361. −3.46372
\(73\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(74\) −97746.0 −2.07501
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 80168.0 1.44522 0.722609 − 0.691257i \(-0.242943\pi\)
0.722609 + 0.691257i \(0.242943\pi\)
\(80\) 0 0
\(81\) 59049.0 1.00000
\(82\) 0 0
\(83\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 129228. 1.88413
\(87\) 0 0
\(88\) −47652.0 −0.655956
\(89\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −440728. −5.42877
\(93\) 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
\(99\) 18468.0 0.189379
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 49.6.a.b.1.1 ✓ 1
3.2 odd 2 441.6.a.a.1.1 1
4.3 odd 2 784.6.a.g.1.1 1
7.2 even 3 49.6.c.a.18.1 2
7.3 odd 6 49.6.c.a.30.1 2
7.4 even 3 49.6.c.a.30.1 2
7.5 odd 6 49.6.c.a.18.1 2
7.6 odd 2 CM 49.6.a.b.1.1 ✓ 1
21.20 even 2 441.6.a.a.1.1 1
28.27 even 2 784.6.a.g.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
49.6.a.b.1.1 ✓ 1 1.1 even 1 trivial
49.6.a.b.1.1 ✓ 1 7.6 odd 2 CM
49.6.c.a.18.1 2 7.2 even 3
49.6.c.a.18.1 2 7.5 odd 6
49.6.c.a.30.1 2 7.3 odd 6
49.6.c.a.30.1 2 7.4 even 3
441.6.a.a.1.1 1 3.2 odd 2
441.6.a.a.1.1 1 21.20 even 2
784.6.a.g.1.1 1 4.3 odd 2
784.6.a.g.1.1 1 28.27 even 2