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Properties

Label 98.8.c.m.67.1
Level $98$
Weight $8$
Character 98.67
Analytic conductor $30.614$
Analytic rank $0$
Dimension $4$
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] = [98,8,Mod(67,98)] 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("98.67"); S:= CuspForms(chi, 8); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(98, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([4])) N = Newforms(chi, 8, names="a")
 
Level: \( N \) \(=\) \( 98 = 2 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 98.c (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,16,56,-128,-14,896,0,-2048,908] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(30.6137324974\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{949})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} + 238x^{2} + 237x + 56169 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 14)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 67.1
Root \(-7.45146 + 12.9063i\) of defining polynomial
Character \(\chi\) \(=\) 98.67
Dual form 98.8.c.m.79.1

$q$-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+(4.00000 + 6.92820i) q^{2} +(-1.40292 + 2.42993i) q^{3} +(-32.0000 + 55.4256i) q^{4} +(-219.141 - 379.563i) q^{5} -22.4467 q^{6} -512.000 q^{8} +(1089.56 + 1887.18i) q^{9} +(1753.13 - 3036.51i) q^{10} +(2740.43 - 4746.57i) q^{11} +(-89.7870 - 155.516i) q^{12} -4006.54 q^{13} +1229.75 q^{15} +(-2048.00 - 3547.24i) q^{16} +(-14011.5 + 24268.6i) q^{17} +(-8716.51 + 15097.4i) q^{18} +(11920.6 + 20647.0i) q^{19} +28050.0 q^{20} +43846.9 q^{22} +(36877.3 + 63873.3i) q^{23} +(718.296 - 1244.13i) q^{24} +(-56983.0 + 98697.4i) q^{25} +(-16026.2 - 27758.1i) q^{26} -12250.7 q^{27} -98721.3 q^{29} +(4919.00 + 8519.96i) q^{30} +(-23743.4 + 41124.8i) q^{31} +(16384.0 - 28377.9i) q^{32} +(7689.23 + 13318.1i) q^{33} -224184. q^{34} -139464. q^{36} +(50031.2 + 86656.7i) q^{37} +(-95364.6 + 165176. i) q^{38} +(5620.87 - 9735.63i) q^{39} +(112200. + 194336. i) q^{40} -489123. q^{41} +299600. q^{43} +(175388. + 303781. i) q^{44} +(477536. - 827116. i) q^{45} +(-295018. + 510986. i) q^{46} +(481369. + 833756. i) q^{47} +11492.7 q^{48} -911728. q^{50} +(-39314.1 - 68093.9i) q^{51} +(128209. - 222065. i) q^{52} +(-918933. + 1.59164e6i) q^{53} +(-49002.7 - 84875.1i) q^{54} -2.40216e6 q^{55} -66894.5 q^{57} +(-394885. - 683961. i) q^{58} +(-7255.29 + 12566.5i) q^{59} +(-39352.0 + 68159.7i) q^{60} +(1.01469e6 + 1.75749e6i) q^{61} -379895. q^{62} +262144. q^{64} +(877997. + 1.52074e6i) q^{65} +(-61513.8 + 106545. i) q^{66} +(1.48449e6 - 2.57121e6i) q^{67} +(-896736. - 1.55319e6i) q^{68} -206944. q^{69} -4.34296e6 q^{71} +(-557857. - 966236. i) q^{72} +(750529. - 1.29995e6i) q^{73} +(-400250. + 693253. i) q^{74} +(-159885. - 276929. i) q^{75} -1.52583e6 q^{76} +89933.9 q^{78} +(886182. + 1.53491e6i) q^{79} +(-897601. + 1.55469e6i) q^{80} +(-2.36569e6 + 4.09749e6i) q^{81} +(-1.95649e6 - 3.38874e6i) q^{82} +1.57509e6 q^{83} +1.22820e7 q^{85} +(1.19840e6 + 2.07569e6i) q^{86} +(138498. - 239886. i) q^{87} +(-1.40310e6 + 2.43024e6i) q^{88} +(4.39727e6 + 7.61629e6i) q^{89} +7.64057e6 q^{90} -4.72029e6 q^{92} +(-66620.4 - 115390. i) q^{93} +(-3.85096e6 + 6.67005e6i) q^{94} +(5.22457e6 - 9.04922e6i) q^{95} +(45970.9 + 79624.0i) q^{96} +1.03493e7 q^{97} +1.19435e7 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 16 q^{2} + 56 q^{3} - 128 q^{4} - 14 q^{5} + 896 q^{6} - 2048 q^{8} + 908 q^{9} + 112 q^{10} - 2408 q^{11} + 3584 q^{12} - 21448 q^{13} + 52360 q^{15} - 8192 q^{16} - 35098 q^{17} - 7264 q^{18} - 2408 q^{19}+ \cdots + 43942528 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/98\mathbb{Z}\right)^\times\).

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

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 4.00000 + 6.92820i 0.353553 + 0.612372i
\(3\) −1.40292 + 2.42993i −0.0299992 + 0.0519601i −0.880635 − 0.473795i \(-0.842884\pi\)
0.850636 + 0.525755i \(0.176217\pi\)
\(4\) −32.0000 + 55.4256i −0.250000 + 0.433013i
\(5\) −219.141 − 379.563i −0.784022 − 1.35797i −0.929581 − 0.368617i \(-0.879831\pi\)
0.145559 − 0.989350i \(-0.453502\pi\)
\(6\) −22.4467 −0.0424252
\(7\) 0 0
\(8\) −512.000 −0.353553
\(9\) 1089.56 + 1887.18i 0.498200 + 0.862908i
\(10\) 1753.13 − 3036.51i 0.554388 − 0.960227i
\(11\) 2740.43 − 4746.57i 0.620790 − 1.07524i −0.368549 − 0.929609i \(-0.620145\pi\)
0.989339 − 0.145632i \(-0.0465215\pi\)
\(12\) −89.7870 − 155.516i −0.0149996 − 0.0259800i
\(13\) −4006.54 −0.505787 −0.252894 − 0.967494i \(-0.581382\pi\)
−0.252894 + 0.967494i \(0.581382\pi\)
\(14\) 0 0
\(15\) 1229.75 0.0940800
\(16\) −2048.00 − 3547.24i −0.125000 − 0.216506i
\(17\) −14011.5 + 24268.6i −0.691693 + 1.19805i 0.279590 + 0.960119i \(0.409801\pi\)
−0.971283 + 0.237927i \(0.923532\pi\)
\(18\) −8716.51 + 15097.4i −0.352281 + 0.610168i
\(19\) 11920.6 + 20647.0i 0.398712 + 0.690590i 0.993567 − 0.113243i \(-0.0361239\pi\)
−0.594855 + 0.803833i \(0.702791\pi\)
\(20\) 28050.0 0.784022
\(21\) 0 0
\(22\) 43846.9 0.877930
\(23\) 36877.3 + 63873.3i 0.631992 + 1.09464i 0.987144 + 0.159833i \(0.0510956\pi\)
−0.355152 + 0.934808i \(0.615571\pi\)
\(24\) 718.296 − 1244.13i 0.0106063 − 0.0183707i
\(25\) −56983.0 + 98697.4i −0.729382 + 1.26333i
\(26\) −16026.2 − 27758.1i −0.178823 − 0.309730i
\(27\) −12250.7 −0.119781
\(28\) 0 0
\(29\) −98721.3 −0.751653 −0.375827 − 0.926690i \(-0.622641\pi\)
−0.375827 + 0.926690i \(0.622641\pi\)
\(30\) 4919.00 + 8519.96i 0.0332623 + 0.0576120i
\(31\) −23743.4 + 41124.8i −0.143145 + 0.247935i −0.928680 − 0.370883i \(-0.879055\pi\)
0.785534 + 0.618818i \(0.212388\pi\)
\(32\) 16384.0 − 28377.9i 0.0883883 − 0.153093i
\(33\) 7689.23 + 13318.1i 0.0372464 + 0.0645126i
\(34\) −224184. −0.978201
\(35\) 0 0
\(36\) −139464. −0.498200
\(37\) 50031.2 + 86656.7i 0.162381 + 0.281252i 0.935722 − 0.352738i \(-0.114749\pi\)
−0.773341 + 0.633990i \(0.781416\pi\)
\(38\) −95364.6 + 165176.i −0.281932 + 0.488321i
\(39\) 5620.87 − 9735.63i 0.0151732 − 0.0262808i
\(40\) 112200. + 194336.i 0.277194 + 0.480114i
\(41\) −489123. −1.10834 −0.554172 − 0.832402i \(-0.686965\pi\)
−0.554172 + 0.832402i \(0.686965\pi\)
\(42\) 0 0
\(43\) 299600. 0.574649 0.287324 − 0.957833i \(-0.407234\pi\)
0.287324 + 0.957833i \(0.407234\pi\)
\(44\) 175388. + 303781.i 0.310395 + 0.537620i
\(45\) 477536. − 827116.i 0.781200 − 1.35308i
\(46\) −295018. + 510986.i −0.446886 + 0.774029i
\(47\) 481369. + 833756.i 0.676295 + 1.17138i 0.976089 + 0.217373i \(0.0697487\pi\)
−0.299794 + 0.954004i \(0.596918\pi\)
\(48\) 11492.7 0.0149996
\(49\) 0 0
\(50\) −911728. −1.03150
\(51\) −39314.1 − 68093.9i −0.0415004 − 0.0718808i
\(52\) 128209. − 222065.i 0.126447 − 0.219012i
\(53\) −918933. + 1.59164e6i −0.847849 + 1.46852i 0.0352758 + 0.999378i \(0.488769\pi\)
−0.883124 + 0.469139i \(0.844564\pi\)
\(54\) −49002.7 − 84875.1i −0.0423489 − 0.0733504i
\(55\) −2.40216e6 −1.94685
\(56\) 0 0
\(57\) −66894.5 −0.0478441
\(58\) −394885. − 683961.i −0.265750 − 0.460292i
\(59\) −7255.29 + 12566.5i −0.00459910 + 0.00796587i −0.868316 − 0.496012i \(-0.834797\pi\)
0.863717 + 0.503978i \(0.168131\pi\)
\(60\) −39352.0 + 68159.7i −0.0235200 + 0.0407379i
\(61\) 1.01469e6 + 1.75749e6i 0.572370 + 0.991374i 0.996322 + 0.0856896i \(0.0273093\pi\)
−0.423952 + 0.905685i \(0.639357\pi\)
\(62\) −379895. −0.202438
\(63\) 0 0
\(64\) 262144. 0.125000
\(65\) 877997. + 1.52074e6i 0.396549 + 0.686842i
\(66\) −61513.8 + 106545.i −0.0263372 + 0.0456173i
\(67\) 1.48449e6 − 2.57121e6i 0.602997 − 1.04442i −0.389368 − 0.921082i \(-0.627306\pi\)
0.992365 − 0.123339i \(-0.0393602\pi\)
\(68\) −896736. − 1.55319e6i −0.345846 − 0.599023i
\(69\) −206944. −0.0758369
\(70\) 0 0
\(71\) −4.34296e6 −1.44006 −0.720031 − 0.693942i \(-0.755872\pi\)
−0.720031 + 0.693942i \(0.755872\pi\)
\(72\) −557857. − 966236.i −0.176140 − 0.305084i
\(73\) 750529. − 1.29995e6i 0.225807 − 0.391109i −0.730754 − 0.682641i \(-0.760831\pi\)
0.956561 + 0.291531i \(0.0941648\pi\)
\(74\) −400250. + 693253.i −0.114821 + 0.198875i
\(75\) −159885. − 276929.i −0.0437617 − 0.0757975i
\(76\) −1.52583e6 −0.398712
\(77\) 0 0
\(78\) 89933.9 0.0214581
\(79\) 886182. + 1.53491e6i 0.202222 + 0.350259i 0.949244 − 0.314541i \(-0.101850\pi\)
−0.747022 + 0.664799i \(0.768517\pi\)
\(80\) −897601. + 1.55469e6i −0.196006 + 0.339492i
\(81\) −2.36569e6 + 4.09749e6i −0.494607 + 0.856684i
\(82\) −1.95649e6 − 3.38874e6i −0.391859 − 0.678720i
\(83\) 1.57509e6 0.302366 0.151183 − 0.988506i \(-0.451692\pi\)
0.151183 + 0.988506i \(0.451692\pi\)
\(84\) 0 0
\(85\) 1.22820e7 2.16921
\(86\) 1.19840e6 + 2.07569e6i 0.203169 + 0.351899i
\(87\) 138498. − 239886.i 0.0225490 − 0.0390560i
\(88\) −1.40310e6 + 2.43024e6i −0.219483 + 0.380155i
\(89\) 4.39727e6 + 7.61629e6i 0.661177 + 1.14519i 0.980307 + 0.197481i \(0.0632761\pi\)
−0.319130 + 0.947711i \(0.603391\pi\)
\(90\) 7.64057e6 1.10478
\(91\) 0 0
\(92\) −4.72029e6 −0.631992
\(93\) −66620.4 − 115390.i −0.00858849 − 0.0148757i
\(94\) −3.85096e6 + 6.67005e6i −0.478213 + 0.828288i
\(95\) 5.22457e6 − 9.04922e6i 0.625199 − 1.08288i
\(96\) 45970.9 + 79624.0i 0.00530315 + 0.00918533i
\(97\) 1.03493e7 1.15135 0.575676 − 0.817678i \(-0.304739\pi\)
0.575676 + 0.817678i \(0.304739\pi\)
\(98\) 0 0
\(99\) 1.19435e7 1.23711
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 98.8.c.m.67.1 4
7.2 even 3 inner 98.8.c.m.79.1 4
7.3 odd 6 98.8.a.f.1.1 2
7.4 even 3 98.8.a.d.1.2 2
7.5 odd 6 14.8.c.b.9.2 ✓ 4
7.6 odd 2 14.8.c.b.11.2 yes 4
21.5 even 6 126.8.g.d.37.1 4
21.20 even 2 126.8.g.d.109.1 4
28.19 even 6 112.8.i.b.65.1 4
28.27 even 2 112.8.i.b.81.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
14.8.c.b.9.2 ✓ 4 7.5 odd 6
14.8.c.b.11.2 yes 4 7.6 odd 2
98.8.a.d.1.2 2 7.4 even 3
98.8.a.f.1.1 2 7.3 odd 6
98.8.c.m.67.1 4 1.1 even 1 trivial
98.8.c.m.79.1 4 7.2 even 3 inner
112.8.i.b.65.1 4 28.19 even 6
112.8.i.b.81.1 4 28.27 even 2
126.8.g.d.37.1 4 21.5 even 6
126.8.g.d.109.1 4 21.20 even 2