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Properties

Label 49.5.d.a.19.1
Level $49$
Weight $5$
Character 49.19
Analytic conductor $5.065$
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] = [49,5,Mod(19,49)] 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("49.19"); S:= CuspForms(chi, 5); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(49, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([5])) N = Newforms(chi, 5, names="a")
 
Level: \( N \) \(=\) \( 49 = 7^{2} \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 49.d (of order \(6\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(5.06512819111\)
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: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 7)
Sato-Tate group: $\mathrm{U}(1)[D_{6}]$

Embedding invariants

Embedding label 19.1
Root \(0.500000 - 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 49.19
Dual form 49.5.d.a.31.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.500000 + 0.866025i) q^{2} +(7.50000 + 12.9904i) q^{4} -31.0000 q^{8} +(-40.5000 + 70.1481i) q^{9} +(103.000 + 178.401i) q^{11} +(-104.500 + 180.999i) q^{16} +(-40.5000 - 70.1481i) q^{18} -206.000 q^{22} +(367.000 - 635.663i) q^{23} +(-312.500 - 541.266i) q^{25} +1234.00 q^{29} +(-352.500 - 610.548i) q^{32} -1215.00 q^{36} +(647.000 - 1120.64i) q^{37} -334.000 q^{43} +(-1545.00 + 2676.02i) q^{44} +(367.000 + 635.663i) q^{46} +625.000 q^{50} +(2791.00 + 4834.15i) q^{53} +(-617.000 + 1068.68i) q^{58} -2639.00 q^{64} +(-2473.00 - 4283.36i) q^{67} +2914.00 q^{71} +(1255.50 - 2174.59i) q^{72} +(647.000 + 1120.64i) q^{74} +(1823.00 - 3157.53i) q^{79} +(-3280.50 - 5681.99i) q^{81} +(167.000 - 289.252i) q^{86} +(-3193.00 - 5530.44i) q^{88} +11010.0 q^{92} -16686.0 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{2} + 15 q^{4} - 62 q^{8} - 81 q^{9} + 206 q^{11} - 209 q^{16} - 81 q^{18} - 412 q^{22} + 734 q^{23} - 625 q^{25} + 2468 q^{29} - 705 q^{32} - 2430 q^{36} + 1294 q^{37} - 668 q^{43} - 3090 q^{44}+ \cdots - 33372 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(3\)
\(\chi(n)\) \(e\left(\frac{5}{6}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.500000 + 0.866025i −0.125000 + 0.216506i −0.921733 − 0.387825i \(-0.873226\pi\)
0.796733 + 0.604332i \(0.206560\pi\)
\(3\) 0 0 −0.500000 − 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(4\) 7.50000 + 12.9904i 0.468750 + 0.811899i
\(5\) 0 0 0.500000 − 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) −31.0000 −0.484375
\(9\) −40.5000 + 70.1481i −0.500000 + 0.866025i
\(10\) 0 0
\(11\) 103.000 + 178.401i 0.851240 + 1.47439i 0.880090 + 0.474807i \(0.157482\pi\)
−0.0288505 + 0.999584i \(0.509185\pi\)
\(12\) 0 0
\(13\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −104.500 + 180.999i −0.408203 + 0.707029i
\(17\) 0 0 −0.500000 − 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(18\) −40.5000 − 70.1481i −0.125000 − 0.216506i
\(19\) 0 0 0.500000 − 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −206.000 −0.425620
\(23\) 367.000 − 635.663i 0.693762 − 1.20163i −0.276834 − 0.960918i \(-0.589285\pi\)
0.970596 − 0.240713i \(-0.0773813\pi\)
\(24\) 0 0
\(25\) −312.500 − 541.266i −0.500000 − 0.866025i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 1234.00 1.46730 0.733650 − 0.679527i \(-0.237815\pi\)
0.733650 + 0.679527i \(0.237815\pi\)
\(30\) 0 0
\(31\) 0 0 −0.500000 − 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(32\) −352.500 − 610.548i −0.344238 − 0.596238i
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) −1215.00 −0.937500
\(37\) 647.000 − 1120.64i 0.472608 − 0.818581i −0.526901 − 0.849927i \(-0.676646\pi\)
0.999509 + 0.0313461i \(0.00997942\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) 0 0
\(43\) −334.000 −0.180638 −0.0903191 − 0.995913i \(-0.528789\pi\)
−0.0903191 + 0.995913i \(0.528789\pi\)
\(44\) −1545.00 + 2676.02i −0.798037 + 1.38224i
\(45\) 0 0
\(46\) 367.000 + 635.663i 0.173440 + 0.300408i
\(47\) 0 0 0.500000 − 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 625.000 0.250000
\(51\) 0 0
\(52\) 0 0
\(53\) 2791.00 + 4834.15i 0.993592 + 1.72095i 0.594679 + 0.803963i \(0.297279\pi\)
0.398913 + 0.916989i \(0.369388\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) −617.000 + 1068.68i −0.183413 + 0.317680i
\(59\) 0 0 −0.500000 − 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(60\) 0 0
\(61\) 0 0 0.500000 − 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) −2639.00 −0.644287
\(65\) 0 0
\(66\) 0 0
\(67\) −2473.00 − 4283.36i −0.550902 − 0.954191i −0.998210 − 0.0598104i \(-0.980950\pi\)
0.447308 − 0.894380i \(-0.352383\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 2914.00 0.578060 0.289030 − 0.957320i \(-0.406667\pi\)
0.289030 + 0.957320i \(0.406667\pi\)
\(72\) 1255.50 − 2174.59i 0.242188 − 0.419481i
\(73\) 0 0 −0.500000 − 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(74\) 647.000 + 1120.64i 0.118152 + 0.204645i
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 1823.00 − 3157.53i 0.292101 − 0.505933i −0.682206 − 0.731160i \(-0.738979\pi\)
0.974306 + 0.225227i \(0.0723124\pi\)
\(80\) 0 0
\(81\) −3280.50 − 5681.99i −0.500000 − 0.866025i
\(82\) 0 0
\(83\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 167.000 − 289.252i 0.0225798 − 0.0391093i
\(87\) 0 0
\(88\) −3193.00 − 5530.44i −0.412319 − 0.714158i
\(89\) 0 0 0.500000 − 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 11010.0 1.30080
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(98\) 0 0
\(99\) −16686.0 −1.70248
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 49.5.d.a.19.1 2
7.2 even 3 7.5.b.a.6.1 ✓ 1
7.3 odd 6 inner 49.5.d.a.31.1 2
7.4 even 3 inner 49.5.d.a.31.1 2
7.5 odd 6 7.5.b.a.6.1 ✓ 1
7.6 odd 2 CM 49.5.d.a.19.1 2
21.2 odd 6 63.5.d.a.55.1 1
21.5 even 6 63.5.d.a.55.1 1
28.19 even 6 112.5.c.a.97.1 1
28.23 odd 6 112.5.c.a.97.1 1
35.2 odd 12 175.5.c.a.174.2 2
35.9 even 6 175.5.d.a.76.1 1
35.12 even 12 175.5.c.a.174.2 2
35.19 odd 6 175.5.d.a.76.1 1
35.23 odd 12 175.5.c.a.174.1 2
35.33 even 12 175.5.c.a.174.1 2
56.5 odd 6 448.5.c.b.321.1 1
56.19 even 6 448.5.c.a.321.1 1
56.37 even 6 448.5.c.b.321.1 1
56.51 odd 6 448.5.c.a.321.1 1
84.23 even 6 1008.5.f.a.433.1 1
84.47 odd 6 1008.5.f.a.433.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
7.5.b.a.6.1 ✓ 1 7.2 even 3
7.5.b.a.6.1 ✓ 1 7.5 odd 6
49.5.d.a.19.1 2 1.1 even 1 trivial
49.5.d.a.19.1 2 7.6 odd 2 CM
49.5.d.a.31.1 2 7.3 odd 6 inner
49.5.d.a.31.1 2 7.4 even 3 inner
63.5.d.a.55.1 1 21.2 odd 6
63.5.d.a.55.1 1 21.5 even 6
112.5.c.a.97.1 1 28.19 even 6
112.5.c.a.97.1 1 28.23 odd 6
175.5.c.a.174.1 2 35.23 odd 12
175.5.c.a.174.1 2 35.33 even 12
175.5.c.a.174.2 2 35.2 odd 12
175.5.c.a.174.2 2 35.12 even 12
175.5.d.a.76.1 1 35.9 even 6
175.5.d.a.76.1 1 35.19 odd 6
448.5.c.a.321.1 1 56.19 even 6
448.5.c.a.321.1 1 56.51 odd 6
448.5.c.b.321.1 1 56.5 odd 6
448.5.c.b.321.1 1 56.37 even 6
1008.5.f.a.433.1 1 84.23 even 6
1008.5.f.a.433.1 1 84.47 odd 6