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Properties

Label 98.14.c.f.79.1
Level $98$
Weight $14$
Character 98.79
Analytic conductor $105.086$
Analytic rank $0$
Dimension $2$
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] = [98,14,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, 14); 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, 14, names="a")
 
Level: \( N \) \(=\) \( 98 = 2 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 14 \)
Character orbit: \([\chi]\) \(=\) 98.c (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,64,-1626] 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: \(105.086310373\)
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: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 14)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 79.1
Root \(0.500000 - 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 98.79
Dual form 98.14.c.f.67.1

$q$-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+(32.0000 - 55.4256i) q^{2} +(-813.000 - 1408.16i) q^{3} +(-2048.00 - 3547.24i) q^{4} +(18200.0 - 31523.3i) q^{5} -104064. q^{6} -262144. q^{8} +(-524776. + 908940. i) q^{9} +(-1.16480e6 - 2.01749e6i) q^{10} +(-1.30264e6 - 2.25625e6i) q^{11} +(-3.33005e6 + 5.76781e6i) q^{12} -1.26245e7 q^{13} -5.91864e7 q^{15} +(-8.38861e6 + 1.45295e7i) q^{16} +(6.53762e7 + 1.13235e8i) q^{17} +(3.35857e7 + 5.81721e7i) q^{18} +(1.24718e8 - 2.16018e8i) q^{19} -1.49094e8 q^{20} -1.66738e8 q^{22} +(-2.44527e8 + 4.23533e8i) q^{23} +(2.13123e8 + 3.69140e8i) q^{24} +(-5.21284e7 - 9.02891e7i) q^{25} +(-4.03983e8 + 6.99719e8i) q^{26} -8.85796e8 q^{27} -1.12116e8 q^{29} +(-1.89396e9 + 3.28044e9i) q^{30} +(4.55153e9 + 7.88349e9i) q^{31} +(5.36871e8 + 9.29888e8i) q^{32} +(-2.11810e9 + 3.66866e9i) q^{33} +8.36815e9 q^{34} +4.29897e9 q^{36} +(-9.15408e9 + 1.58553e10i) q^{37} +(-7.98195e9 - 1.38251e10i) q^{38} +(1.02637e10 + 1.77772e10i) q^{39} +(-4.77102e9 + 8.26365e9i) q^{40} +1.30824e10 q^{41} -6.71235e10 q^{43} +(-5.33563e9 + 9.24158e9i) q^{44} +(1.91019e10 + 3.30854e10i) q^{45} +(1.56497e10 + 2.71061e10i) q^{46} +(-5.26200e10 + 9.11405e10i) q^{47} +2.72798e10 q^{48} -6.67244e9 q^{50} +(1.06302e11 - 1.84120e11i) q^{51} +(2.58549e10 + 4.47820e10i) q^{52} +(1.26109e10 + 2.18427e10i) q^{53} +(-2.83455e10 + 4.90958e10i) q^{54} -9.48325e10 q^{55} -4.05583e11 q^{57} +(-3.58771e9 + 6.21410e9i) q^{58} +(1.38387e11 + 2.39694e11i) q^{59} +(1.21214e11 + 2.09948e11i) q^{60} +(-3.79694e11 + 6.57650e11i) q^{61} +5.82596e11 q^{62} +6.87195e10 q^{64} +(-2.29765e11 + 3.97965e11i) q^{65} +(1.35558e11 + 2.34794e11i) q^{66} +(-5.19832e11 - 9.00376e11i) q^{67} +(2.67781e11 - 4.63810e11i) q^{68} +7.95202e11 q^{69} +1.81709e12 q^{71} +(1.37567e11 - 2.38273e11i) q^{72} +(-2.00171e11 - 3.46707e11i) q^{73} +(5.85861e11 + 1.01474e12i) q^{74} +(-8.47608e10 + 1.46810e11i) q^{75} -1.02169e12 q^{76} +1.31375e12 q^{78} +(1.79890e12 - 3.11578e12i) q^{79} +(3.05345e11 + 5.28874e11i) q^{80} +(1.55682e12 + 2.69648e12i) q^{81} +(4.18636e11 - 7.25099e11i) q^{82} -1.30903e12 q^{83} +4.75939e12 q^{85} +(-2.14795e12 + 3.72036e12i) q^{86} +(9.11502e10 + 1.57877e11i) q^{87} +(3.41480e11 + 5.91461e11i) q^{88} +(-8.26644e11 + 1.43179e12i) q^{89} +2.44504e12 q^{90} +2.00317e12 q^{92} +(7.40079e12 - 1.28186e13i) q^{93} +(3.36768e12 + 5.83299e12i) q^{94} +(-4.53974e12 - 7.86305e12i) q^{95} +(8.72952e11 - 1.51200e12i) q^{96} -1.27369e13 q^{97} +2.73439e12 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 64 q^{2} - 1626 q^{3} - 4096 q^{4} + 36400 q^{5} - 208128 q^{6} - 524288 q^{8} - 1049553 q^{9} - 2329600 q^{10} - 2605288 q^{11} - 6660096 q^{12} - 25248936 q^{13} - 118372800 q^{15} - 16777216 q^{16}+ \cdots + 5468775672528 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(3\)
\(\chi(n)\) \(e\left(\frac{1}{3}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 32.0000 − 55.4256i 0.353553 − 0.612372i
\(3\) −813.000 − 1408.16i −0.643876 − 1.11523i −0.984560 − 0.175048i \(-0.943992\pi\)
0.340684 − 0.940178i \(-0.389342\pi\)
\(4\) −2048.00 − 3547.24i −0.250000 − 0.433013i
\(5\) 18200.0 − 31523.3i 0.520914 − 0.902250i −0.478790 − 0.877930i \(-0.658924\pi\)
0.999704 − 0.0243206i \(-0.00774226\pi\)
\(6\) −104064. −0.910578
\(7\) 0 0
\(8\) −262144. −0.353553
\(9\) −524776. + 908940.i −0.329153 + 0.570110i
\(10\) −1.16480e6 − 2.01749e6i −0.368342 − 0.637987i
\(11\) −1.30264e6 − 2.25625e6i −0.221704 − 0.384002i 0.733622 − 0.679558i \(-0.237828\pi\)
−0.955325 + 0.295556i \(0.904495\pi\)
\(12\) −3.33005e6 + 5.76781e6i −0.321938 + 0.557613i
\(13\) −1.26245e7 −0.725406 −0.362703 − 0.931905i \(-0.618146\pi\)
−0.362703 + 0.931905i \(0.618146\pi\)
\(14\) 0 0
\(15\) −5.91864e7 −1.34162
\(16\) −8.38861e6 + 1.45295e7i −0.125000 + 0.216506i
\(17\) 6.53762e7 + 1.13235e8i 0.656903 + 1.13779i 0.981413 + 0.191908i \(0.0614674\pi\)
−0.324509 + 0.945882i \(0.605199\pi\)
\(18\) 3.35857e7 + 5.81721e7i 0.232746 + 0.403129i
\(19\) 1.24718e8 − 2.16018e8i 0.608178 − 1.05340i −0.383362 − 0.923598i \(-0.625234\pi\)
0.991541 − 0.129797i \(-0.0414327\pi\)
\(20\) −1.49094e8 −0.520914
\(21\) 0 0
\(22\) −1.66738e8 −0.313537
\(23\) −2.44527e8 + 4.23533e8i −0.344426 + 0.596563i −0.985249 − 0.171125i \(-0.945260\pi\)
0.640823 + 0.767688i \(0.278593\pi\)
\(24\) 2.13123e8 + 3.69140e8i 0.227645 + 0.394292i
\(25\) −5.21284e7 − 9.02891e7i −0.0427036 − 0.0739648i
\(26\) −4.03983e8 + 6.99719e8i −0.256470 + 0.444219i
\(27\) −8.85796e8 −0.440017
\(28\) 0 0
\(29\) −1.12116e8 −0.0350010 −0.0175005 − 0.999847i \(-0.505571\pi\)
−0.0175005 + 0.999847i \(0.505571\pi\)
\(30\) −1.89396e9 + 3.28044e9i −0.474333 + 0.821570i
\(31\) 4.55153e9 + 7.88349e9i 0.921100 + 1.59539i 0.797716 + 0.603033i \(0.206041\pi\)
0.123384 + 0.992359i \(0.460625\pi\)
\(32\) 5.36871e8 + 9.29888e8i 0.0883883 + 0.153093i
\(33\) −2.11810e9 + 3.66866e9i −0.285500 + 0.494500i
\(34\) 8.36815e9 0.929002
\(35\) 0 0
\(36\) 4.29897e9 0.329153
\(37\) −9.15408e9 + 1.58553e10i −0.586548 + 1.01593i 0.408132 + 0.912923i \(0.366180\pi\)
−0.994680 + 0.103009i \(0.967153\pi\)
\(38\) −7.98195e9 − 1.38251e10i −0.430047 − 0.744863i
\(39\) 1.02637e10 + 1.77772e10i 0.467072 + 0.808992i
\(40\) −4.77102e9 + 8.26365e9i −0.184171 + 0.318994i
\(41\) 1.30824e10 0.430122 0.215061 − 0.976601i \(-0.431005\pi\)
0.215061 + 0.976601i \(0.431005\pi\)
\(42\) 0 0
\(43\) −6.71235e10 −1.61931 −0.809654 − 0.586908i \(-0.800345\pi\)
−0.809654 + 0.586908i \(0.800345\pi\)
\(44\) −5.33563e9 + 9.24158e9i −0.110852 + 0.192001i
\(45\) 1.91019e10 + 3.30854e10i 0.342921 + 0.593957i
\(46\) 1.56497e10 + 2.71061e10i 0.243546 + 0.421834i
\(47\) −5.26200e10 + 9.11405e10i −0.712057 + 1.23332i 0.252027 + 0.967720i \(0.418903\pi\)
−0.964084 + 0.265598i \(0.914430\pi\)
\(48\) 2.72798e10 0.321938
\(49\) 0 0
\(50\) −6.67244e9 −0.0603920
\(51\) 1.06302e11 − 1.84120e11i 0.845929 − 1.46519i
\(52\) 2.58549e10 + 4.47820e10i 0.181352 + 0.314110i
\(53\) 1.26109e10 + 2.18427e10i 0.0781541 + 0.135367i 0.902454 − 0.430787i \(-0.141764\pi\)
−0.824299 + 0.566154i \(0.808431\pi\)
\(54\) −2.83455e10 + 4.90958e10i −0.155569 + 0.269454i
\(55\) −9.48325e10 −0.461955
\(56\) 0 0
\(57\) −4.05583e11 −1.56637
\(58\) −3.58771e9 + 6.21410e9i −0.0123747 + 0.0214336i
\(59\) 1.38387e11 + 2.39694e11i 0.427128 + 0.739808i 0.996617 − 0.0821918i \(-0.0261920\pi\)
−0.569488 + 0.821999i \(0.692859\pi\)
\(60\) 1.21214e11 + 2.09948e11i 0.335404 + 0.580937i
\(61\) −3.79694e11 + 6.57650e11i −0.943605 + 1.63437i −0.185084 + 0.982723i \(0.559256\pi\)
−0.758521 + 0.651649i \(0.774078\pi\)
\(62\) 5.82596e11 1.30263
\(63\) 0 0
\(64\) 6.87195e10 0.125000
\(65\) −2.29765e11 + 3.97965e11i −0.377875 + 0.654498i
\(66\) 1.35558e11 + 2.34794e11i 0.201879 + 0.349664i
\(67\) −5.19832e11 − 9.00376e11i −0.702065 − 1.21601i −0.967740 − 0.251949i \(-0.918928\pi\)
0.265676 − 0.964062i \(-0.414405\pi\)
\(68\) 2.67781e11 − 4.63810e11i 0.328452 − 0.568895i
\(69\) 7.95202e11 0.887071
\(70\) 0 0
\(71\) 1.81709e12 1.68343 0.841717 − 0.539918i \(-0.181545\pi\)
0.841717 + 0.539918i \(0.181545\pi\)
\(72\) 1.37567e11 − 2.38273e11i 0.116373 − 0.201564i
\(73\) −2.00171e11 − 3.46707e11i −0.154811 − 0.268141i 0.778179 − 0.628043i \(-0.216144\pi\)
−0.932990 + 0.359901i \(0.882810\pi\)
\(74\) 5.85861e11 + 1.01474e12i 0.414752 + 0.718372i
\(75\) −8.47608e10 + 1.46810e11i −0.0549917 + 0.0952484i
\(76\) −1.02169e12 −0.608178
\(77\) 0 0
\(78\) 1.31375e12 0.660539
\(79\) 1.79890e12 − 3.11578e12i 0.832589 − 1.44209i −0.0633891 − 0.997989i \(-0.520191\pi\)
0.895978 − 0.444098i \(-0.146476\pi\)
\(80\) 3.05345e11 + 5.28874e11i 0.130229 + 0.225563i
\(81\) 1.55682e12 + 2.69648e12i 0.612470 + 1.06083i
\(82\) 4.18636e11 − 7.25099e11i 0.152071 − 0.263395i
\(83\) −1.30903e12 −0.439483 −0.219742 − 0.975558i \(-0.570521\pi\)
−0.219742 + 0.975558i \(0.570521\pi\)
\(84\) 0 0
\(85\) 4.75939e12 1.36876
\(86\) −2.14795e12 + 3.72036e12i −0.572512 + 0.991619i
\(87\) 9.11502e10 + 1.57877e11i 0.0225363 + 0.0390340i
\(88\) 3.41480e11 + 5.91461e11i 0.0783842 + 0.135765i
\(89\) −8.26644e11 + 1.43179e12i −0.176313 + 0.305383i −0.940615 − 0.339476i \(-0.889750\pi\)
0.764302 + 0.644858i \(0.223084\pi\)
\(90\) 2.44504e12 0.484964
\(91\) 0 0
\(92\) 2.00317e12 0.344426
\(93\) 7.40079e12 − 1.28186e13i 1.18615 − 2.05447i
\(94\) 3.36768e12 + 5.83299e12i 0.503500 + 0.872088i
\(95\) −4.53974e12 − 7.86305e12i −0.633617 − 1.09746i
\(96\) 8.72952e11 − 1.51200e12i 0.113822 − 0.197146i
\(97\) −1.27369e13 −1.55256 −0.776279 − 0.630390i \(-0.782895\pi\)
−0.776279 + 0.630390i \(0.782895\pi\)
\(98\) 0 0
\(99\) 2.73439e12 0.291898
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 98.14.c.f.79.1 2
7.2 even 3 14.14.a.a.1.1 ✓ 1
7.3 odd 6 98.14.c.g.67.1 2
7.4 even 3 inner 98.14.c.f.67.1 2
7.5 odd 6 98.14.a.a.1.1 1
7.6 odd 2 98.14.c.g.79.1 2
21.2 odd 6 126.14.a.e.1.1 1
28.23 odd 6 112.14.a.a.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
14.14.a.a.1.1 ✓ 1 7.2 even 3
98.14.a.a.1.1 1 7.5 odd 6
98.14.c.f.67.1 2 7.4 even 3 inner
98.14.c.f.79.1 2 1.1 even 1 trivial
98.14.c.g.67.1 2 7.3 odd 6
98.14.c.g.79.1 2 7.6 odd 2
112.14.a.a.1.1 1 28.23 odd 6
126.14.a.e.1.1 1 21.2 odd 6