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Properties

Label 98.14.c.c.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,-64,1026] 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.c.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} +(513.000 + 888.542i) q^{3} +(-2048.00 - 3547.24i) q^{4} +(-2160.00 + 3741.23i) q^{5} -65664.0 q^{6} +262144. q^{8} +(270824. - 469080. i) q^{9} +(-138240. - 239439. i) q^{10} +(4.39366e6 + 7.61004e6i) q^{11} +(2.10125e6 - 3.63947e6i) q^{12} -2.04209e7 q^{13} -4.43232e6 q^{15} +(-8.38861e6 + 1.45295e7i) q^{16} +(-859731. - 1.48910e6i) q^{17} +(1.73327e7 + 3.00211e7i) q^{18} +(5.48515e7 - 9.50055e7i) q^{19} +1.76947e7 q^{20} -5.62388e8 q^{22} +(3.23380e8 - 5.60111e8i) q^{23} +(1.34480e8 + 2.32926e8i) q^{24} +(6.01020e8 + 1.04100e9i) q^{25} +(6.53470e8 - 1.13184e9i) q^{26} +2.19151e9 q^{27} +7.28867e8 q^{29} +(1.41834e8 - 2.45664e8i) q^{30} +(-5.14025e8 - 8.90317e8i) q^{31} +(-5.36871e8 - 9.29888e8i) q^{32} +(-4.50789e9 + 7.80790e9i) q^{33} +1.10046e8 q^{34} -2.21859e9 q^{36} +(-7.11470e9 + 1.23230e10i) q^{37} +(3.51049e9 + 6.08035e9i) q^{38} +(-1.04759e10 - 1.81449e10i) q^{39} +(-5.66231e8 + 9.80741e8i) q^{40} +4.45445e10 q^{41} -5.46898e10 q^{43} +(1.79964e10 - 3.11707e10i) q^{44} +(1.16996e9 + 2.02643e9i) q^{45} +(2.06963e10 + 3.58471e10i) q^{46} +(-2.39342e10 + 4.14552e10i) q^{47} -1.72134e10 q^{48} -7.69306e10 q^{50} +(8.82084e8 - 1.52781e9i) q^{51} +(4.18221e10 + 7.24379e10i) q^{52} +(8.49934e10 + 1.47213e11i) q^{53} +(-7.01282e10 + 1.21466e11i) q^{54} -3.79612e10 q^{55} +1.12555e11 q^{57} +(-2.33238e10 + 4.03979e10i) q^{58} +(1.50383e11 + 2.60471e11i) q^{59} +(9.07739e9 + 1.57225e10i) q^{60} +(-1.84998e11 + 3.20426e11i) q^{61} +6.57951e10 q^{62} +6.87195e10 q^{64} +(4.41092e10 - 7.63994e10i) q^{65} +(-2.88505e11 - 4.99705e11i) q^{66} +(3.93505e11 + 6.81571e11i) q^{67} +(-3.52146e9 + 6.09934e9i) q^{68} +6.63576e11 q^{69} +5.59441e11 q^{71} +(7.09948e10 - 1.22967e11i) q^{72} +(-6.05688e10 - 1.04908e11i) q^{73} +(-4.55341e11 - 7.88673e11i) q^{74} +(-6.16647e11 + 1.06806e12i) q^{75} -4.49343e11 q^{76} +1.34092e12 q^{78} +(-1.45213e11 + 2.51517e11i) q^{79} +(-3.62388e10 - 6.27674e10i) q^{80} +(6.92462e11 + 1.19938e12i) q^{81} +(-1.42542e12 + 2.46890e12i) q^{82} -3.96511e12 q^{83} +7.42808e9 q^{85} +(1.75007e12 - 3.03122e12i) q^{86} +(3.73909e11 + 6.47629e11i) q^{87} +(1.15177e12 + 1.99493e12i) q^{88} +(3.01296e12 - 5.21860e12i) q^{89} -1.49755e11 q^{90} -2.64913e12 q^{92} +(5.27389e11 - 9.13465e11i) q^{93} +(-1.53179e12 - 2.65313e12i) q^{94} +(2.36958e11 + 4.10424e11i) q^{95} +(5.50830e11 - 9.54065e11i) q^{96} +1.13028e13 q^{97} +4.75962e12 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 64 q^{2} + 1026 q^{3} - 4096 q^{4} - 4320 q^{5} - 131328 q^{6} + 524288 q^{8} + 541647 q^{9} - 276480 q^{10} + 8787312 q^{11} + 4202496 q^{12} - 40841864 q^{13} - 8864640 q^{15} - 16777216 q^{16}+ \cdots + 9519242365728 q^{99}+O(q^{100}) \) 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
\(3\) 513.000 + 888.542i 0.406284 + 0.703704i 0.994470 − 0.105022i \(-0.0334912\pi\)
−0.588186 + 0.808725i \(0.700158\pi\)
\(4\) −2048.00 − 3547.24i −0.250000 − 0.433013i
\(5\) −2160.00 + 3741.23i −0.0618228 + 0.107080i −0.895280 − 0.445503i \(-0.853025\pi\)
0.833457 + 0.552584i \(0.186358\pi\)
\(6\) −65664.0 −0.574572
\(7\) 0 0
\(8\) 262144. 0.353553
\(9\) 270824. − 469080.i 0.169867 − 0.294219i
\(10\) −138240. − 239439.i −0.0437153 − 0.0757172i
\(11\) 4.39366e6 + 7.61004e6i 0.747780 + 1.29519i 0.948885 + 0.315623i \(0.102213\pi\)
−0.201105 + 0.979570i \(0.564453\pi\)
\(12\) 2.10125e6 − 3.63947e6i 0.203142 − 0.351852i
\(13\) −2.04209e7 −1.17339 −0.586697 − 0.809807i \(-0.699572\pi\)
−0.586697 + 0.809807i \(0.699572\pi\)
\(14\) 0 0
\(15\) −4.43232e6 −0.100470
\(16\) −8.38861e6 + 1.45295e7i −0.125000 + 0.216506i
\(17\) −859731. − 1.48910e6i −0.00863862 − 0.0149625i 0.861674 − 0.507463i \(-0.169416\pi\)
−0.870312 + 0.492500i \(0.836083\pi\)
\(18\) 1.73327e7 + 3.00211e7i 0.120114 + 0.208044i
\(19\) 5.48515e7 − 9.50055e7i 0.267479 − 0.463287i −0.700731 − 0.713426i \(-0.747143\pi\)
0.968210 + 0.250138i \(0.0804760\pi\)
\(20\) 1.76947e7 0.0618228
\(21\) 0 0
\(22\) −5.62388e8 −1.05752
\(23\) 3.23380e8 − 5.60111e8i 0.455494 − 0.788938i −0.543223 − 0.839589i \(-0.682796\pi\)
0.998716 + 0.0506504i \(0.0161294\pi\)
\(24\) 1.34480e8 + 2.32926e8i 0.143643 + 0.248797i
\(25\) 6.01020e8 + 1.04100e9i 0.492356 + 0.852785i
\(26\) 6.53470e8 − 1.13184e9i 0.414857 − 0.718554i
\(27\) 2.19151e9 1.08862
\(28\) 0 0
\(29\) 7.28867e8 0.227542 0.113771 − 0.993507i \(-0.463707\pi\)
0.113771 + 0.993507i \(0.463707\pi\)
\(30\) 1.41834e8 − 2.45664e8i 0.0355216 − 0.0615253i
\(31\) −5.14025e8 − 8.90317e8i −0.104024 − 0.180175i 0.809315 − 0.587375i \(-0.199838\pi\)
−0.913339 + 0.407200i \(0.866505\pi\)
\(32\) −5.36871e8 − 9.29888e8i −0.0883883 − 0.153093i
\(33\) −4.50789e9 + 7.80790e9i −0.607621 + 1.05243i
\(34\) 1.10046e8 0.0122169
\(35\) 0 0
\(36\) −2.21859e9 −0.169867
\(37\) −7.11470e9 + 1.23230e10i −0.455874 + 0.789597i −0.998738 − 0.0502234i \(-0.984007\pi\)
0.542864 + 0.839821i \(0.317340\pi\)
\(38\) 3.51049e9 + 6.08035e9i 0.189136 + 0.327594i
\(39\) −1.04759e10 − 1.81449e10i −0.476731 − 0.825722i
\(40\) −5.66231e8 + 9.80741e8i −0.0218577 + 0.0378586i
\(41\) 4.45445e10 1.46453 0.732266 − 0.681019i \(-0.238463\pi\)
0.732266 + 0.681019i \(0.238463\pi\)
\(42\) 0 0
\(43\) −5.46898e10 −1.31935 −0.659677 − 0.751549i \(-0.729307\pi\)
−0.659677 + 0.751549i \(0.729307\pi\)
\(44\) 1.79964e10 − 3.11707e10i 0.373890 − 0.647596i
\(45\) 1.16996e9 + 2.02643e9i 0.0210034 + 0.0363789i
\(46\) 2.06963e10 + 3.58471e10i 0.322083 + 0.557864i
\(47\) −2.39342e10 + 4.14552e10i −0.323879 + 0.560974i −0.981285 − 0.192562i \(-0.938320\pi\)
0.657406 + 0.753536i \(0.271654\pi\)
\(48\) −1.72134e10 −0.203142
\(49\) 0 0
\(50\) −7.69306e10 −0.696296
\(51\) 8.82084e8 − 1.52781e9i 0.00701946 − 0.0121581i
\(52\) 4.18221e10 + 7.24379e10i 0.293348 + 0.508094i
\(53\) 8.49934e10 + 1.47213e11i 0.526735 + 0.912332i 0.999515 + 0.0311512i \(0.00991735\pi\)
−0.472780 + 0.881181i \(0.656749\pi\)
\(54\) −7.01282e10 + 1.21466e11i −0.384887 + 0.666644i
\(55\) −3.79612e10 −0.184919
\(56\) 0 0
\(57\) 1.12555e11 0.434689
\(58\) −2.33238e10 + 4.03979e10i −0.0804482 + 0.139340i
\(59\) 1.50383e11 + 2.60471e11i 0.464152 + 0.803934i 0.999163 − 0.0409106i \(-0.0130259\pi\)
−0.535011 + 0.844845i \(0.679693\pi\)
\(60\) 9.07739e9 + 1.57225e10i 0.0251176 + 0.0435049i
\(61\) −1.84998e11 + 3.20426e11i −0.459752 + 0.796313i −0.998948 − 0.0458670i \(-0.985395\pi\)
0.539196 + 0.842180i \(0.318728\pi\)
\(62\) 6.57951e10 0.147112
\(63\) 0 0
\(64\) 6.87195e10 0.125000
\(65\) 4.41092e10 − 7.63994e10i 0.0725425 − 0.125647i
\(66\) −2.88505e11 − 4.99705e11i −0.429653 − 0.744181i
\(67\) 3.93505e11 + 6.81571e11i 0.531453 + 0.920503i 0.999326 + 0.0367074i \(0.0116869\pi\)
−0.467874 + 0.883795i \(0.654980\pi\)
\(68\) −3.52146e9 + 6.09934e9i −0.00431931 + 0.00748127i
\(69\) 6.63576e11 0.740238
\(70\) 0 0
\(71\) 5.59441e11 0.518293 0.259147 − 0.965838i \(-0.416559\pi\)
0.259147 + 0.965838i \(0.416559\pi\)
\(72\) 7.09948e10 − 1.22967e11i 0.0600572 − 0.104022i
\(73\) −6.05688e10 − 1.04908e11i −0.0468436 − 0.0811355i 0.841653 − 0.540019i \(-0.181583\pi\)
−0.888497 + 0.458883i \(0.848250\pi\)
\(74\) −4.55341e11 − 7.88673e11i −0.322352 − 0.558330i
\(75\) −6.16647e11 + 1.06806e12i −0.400072 + 0.692945i
\(76\) −4.49343e11 −0.267479
\(77\) 0 0
\(78\) 1.34092e12 0.674199
\(79\) −1.45213e11 + 2.51517e11i −0.0672095 + 0.116410i −0.897672 − 0.440664i \(-0.854743\pi\)
0.830462 + 0.557075i \(0.188076\pi\)
\(80\) −3.62388e10 − 6.27674e10i −0.0154557 − 0.0267701i
\(81\) 6.92462e11 + 1.19938e12i 0.272423 + 0.471850i
\(82\) −1.42542e12 + 2.46890e12i −0.517790 + 0.896839i
\(83\) −3.96511e12 −1.33121 −0.665606 − 0.746303i \(-0.731827\pi\)
−0.665606 + 0.746303i \(0.731827\pi\)
\(84\) 0 0
\(85\) 7.42808e9 0.00213626
\(86\) 1.75007e12 − 3.03122e12i 0.466462 − 0.807937i
\(87\) 3.73909e11 + 6.47629e11i 0.0924465 + 0.160122i
\(88\) 1.15177e12 + 1.99493e12i 0.264380 + 0.457920i
\(89\) 3.01296e12 − 5.21860e12i 0.642626 − 1.11306i −0.342218 − 0.939620i \(-0.611178\pi\)
0.984844 − 0.173440i \(-0.0554884\pi\)
\(90\) −1.49755e11 −0.0297032
\(91\) 0 0
\(92\) −2.64913e12 −0.455494
\(93\) 5.27389e11 − 9.13465e11i 0.0845263 − 0.146404i
\(94\) −1.53179e12 − 2.65313e12i −0.229017 − 0.396669i
\(95\) 2.36958e11 + 4.10424e11i 0.0330726 + 0.0572835i
\(96\) 5.50830e11 − 9.54065e11i 0.0718215 − 0.124398i
\(97\) 1.13028e13 1.37775 0.688875 − 0.724880i \(-0.258105\pi\)
0.688875 + 0.724880i \(0.258105\pi\)
\(98\) 0 0
\(99\) 4.75962e12 0.508093
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.c.79.1 2
7.2 even 3 14.14.a.b.1.1 ✓ 1
7.3 odd 6 98.14.c.b.67.1 2
7.4 even 3 inner 98.14.c.c.67.1 2
7.5 odd 6 98.14.a.d.1.1 1
7.6 odd 2 98.14.c.b.79.1 2
21.2 odd 6 126.14.a.a.1.1 1
28.23 odd 6 112.14.a.b.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
14.14.a.b.1.1 ✓ 1 7.2 even 3
98.14.a.d.1.1 1 7.5 odd 6
98.14.c.b.67.1 2 7.3 odd 6
98.14.c.b.79.1 2 7.6 odd 2
98.14.c.c.67.1 2 7.4 even 3 inner
98.14.c.c.79.1 2 1.1 even 1 trivial
112.14.a.b.1.1 1 28.23 odd 6
126.14.a.a.1.1 1 21.2 odd 6