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Properties

Label 98.7.f.a.13.17
Level $98$
Weight $7$
Character 98.13
Analytic conductor $22.545$
Analytic rank $0$
Dimension $168$
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] = [98,7,Mod(13,98)] 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("98.13"); S:= CuspForms(chi, 7); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(98, base_ring=CyclotomicField(14)) chi = DirichletCharacter(H, H._module([11])) N = Newforms(chi, 7, names="a")
 
Level: \( N \) \(=\) \( 98 = 2 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 98.f (of order \(14\), degree \(6\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(22.5453001947\)
Analytic rank: \(0\)
Dimension: \(168\)
Relative dimension: \(28\) over \(\Q(\zeta_{14})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{14}]$

Embedding invariants

Embedding label 13.17
Character \(\chi\) \(=\) 98.13
Dual form 98.7.f.a.83.17

$q$-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+(1.25877 + 5.51503i) q^{2} +(-22.8730 + 18.2406i) q^{3} +(-28.8310 + 13.8843i) q^{4} +(89.6887 - 71.5244i) q^{5} +(-129.389 - 103.184i) q^{6} +(221.241 + 262.110i) q^{7} +(-112.864 - 141.527i) q^{8} +(28.2365 - 123.712i) q^{9} +(507.356 + 404.603i) q^{10} +(423.340 + 1854.77i) q^{11} +(406.194 - 843.470i) q^{12} +(-1529.52 + 349.102i) q^{13} +(-1167.05 + 1550.08i) q^{14} +(-746.802 + 3271.95i) q^{15} +(638.454 - 800.595i) q^{16} +(-945.442 + 1963.23i) q^{17} +717.818 q^{18} -726.021i q^{19} +(-1592.75 + 3307.38i) q^{20} +(-9841.48 - 1959.67i) q^{21} +(-9696.23 + 4669.46i) q^{22} +(8368.08 - 4029.86i) q^{23} +(5163.06 + 1178.44i) q^{24} +(-548.557 + 2403.39i) q^{25} +(-3850.61 - 7995.88i) q^{26} +(-7642.87 - 15870.6i) q^{27} +(-10017.8 - 4485.12i) q^{28} +(-28441.0 - 13696.5i) q^{29} -18985.0 q^{30} -21826.8i q^{31} +(5218.97 + 2513.32i) q^{32} +(-43515.2 - 34702.2i) q^{33} +(-12017.4 - 2742.88i) q^{34} +(38590.0 + 7684.17i) q^{35} +(903.567 + 3958.79i) q^{36} +(11206.7 + 5396.86i) q^{37} +(4004.03 - 913.893i) q^{38} +(28616.8 - 35884.3i) q^{39} +(-20245.2 - 4620.84i) q^{40} +(-42261.8 + 33702.7i) q^{41} +(-1580.54 - 56742.8i) q^{42} +(-28339.6 + 35536.8i) q^{43} +(-37957.5 - 47597.2i) q^{44} +(-6315.94 - 13115.2i) q^{45} +(32758.2 + 41077.5i) q^{46} +(-55565.4 + 12682.4i) q^{47} +29957.8i q^{48} +(-19754.0 + 115979. i) q^{49} -13945.2 q^{50} +(-14185.4 - 62150.4i) q^{51} +(39250.4 - 31301.2i) q^{52} +(-52611.9 + 25336.6i) q^{53} +(77906.1 - 62128.0i) q^{54} +(170630. + 136073. i) q^{55} +(12125.4 - 60894.1i) q^{56} +(13243.1 + 16606.3i) q^{57} +(39735.8 - 174094. i) q^{58} +(-274383. - 218813. i) q^{59} +(-23897.7 - 104703. i) q^{60} +(-31421.0 + 65246.3i) q^{61} +(120375. - 27474.9i) q^{62} +(38673.2 - 19969.1i) q^{63} +(-7291.57 + 31946.4i) q^{64} +(-112211. + 140708. i) q^{65} +(136608. - 283669. i) q^{66} +45489.4 q^{67} -69728.7i q^{68} +(-117896. + 244814. i) q^{69} +(6197.54 + 222498. i) q^{70} +(-465423. + 224136. i) q^{71} +(-20695.4 + 9966.39i) q^{72} +(-236081. - 53884.0i) q^{73} +(-15657.2 + 68598.7i) q^{74} +(-31292.1 - 64978.6i) q^{75} +(10080.3 + 20931.9i) q^{76} +(-392494. + 521313. i) q^{77} +(233925. + 112652. i) q^{78} +365705. q^{79} -117469. i q^{80} +(547649. + 263734. i) q^{81} +(-239069. - 190651. i) q^{82} +(548050. + 125089. i) q^{83} +(310948. - 80142.7i) q^{84} +(55623.3 + 243702. i) q^{85} +(-231659. - 111561. i) q^{86} +(900364. - 205502. i) q^{87} +(214720. - 269250. i) q^{88} +(975174. + 222577. i) q^{89} +(64380.2 - 51341.5i) q^{90} +(-429894. - 323665. i) q^{91} +(-185309. + 232370. i) q^{92} +(398134. + 499245. i) q^{93} +(-139888. - 290481. i) q^{94} +(-51928.2 - 65115.9i) q^{95} +(-165218. + 37709.9i) q^{96} +1.44607e6i q^{97} +(-664491. + 37046.8i) q^{98} +241411. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 168 q - 896 q^{4} + 784 q^{6} - 308 q^{7} + 9884 q^{9} - 3976 q^{11} - 2688 q^{14} - 3948 q^{15} - 28672 q^{16} + 6272 q^{17} + 11200 q^{18} - 25088 q^{20} - 19488 q^{21} + 16800 q^{22} + 81536 q^{23} + 111524 q^{25}+ \cdots - 7151816 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(3\)
\(\chi(n)\) \(e\left(\frac{11}{14}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.25877 + 5.51503i 0.157346 + 0.689378i
\(3\) −22.8730 + 18.2406i −0.847148 + 0.675578i −0.947633 − 0.319361i \(-0.896532\pi\)
0.100485 + 0.994939i \(0.467960\pi\)
\(4\) −28.8310 + 13.8843i −0.450484 + 0.216942i
\(5\) 89.6887 − 71.5244i 0.717510 − 0.572195i −0.195220 − 0.980760i \(-0.562542\pi\)
0.912729 + 0.408565i \(0.133971\pi\)
\(6\) −129.389 − 103.184i −0.599024 − 0.477706i
\(7\) 221.241 + 262.110i 0.645017 + 0.764168i
\(8\) −112.864 − 141.527i −0.220437 − 0.276419i
\(9\) 28.2365 − 123.712i 0.0387332 − 0.169701i
\(10\) 507.356 + 404.603i 0.507356 + 0.404603i
\(11\) 423.340 + 1854.77i 0.318061 + 1.39352i 0.840948 + 0.541115i \(0.181998\pi\)
−0.522887 + 0.852402i \(0.675145\pi\)
\(12\) 406.194 − 843.470i 0.235066 − 0.488119i
\(13\) −1529.52 + 349.102i −0.696184 + 0.158899i −0.555946 − 0.831218i \(-0.687644\pi\)
−0.140238 + 0.990118i \(0.544787\pi\)
\(14\) −1167.05 + 1550.08i −0.425310 + 0.564900i
\(15\) −746.802 + 3271.95i −0.221275 + 0.969468i
\(16\) 638.454 − 800.595i 0.155872 − 0.195458i
\(17\) −945.442 + 1963.23i −0.192437 + 0.399599i −0.974754 − 0.223281i \(-0.928323\pi\)
0.782317 + 0.622880i \(0.214038\pi\)
\(18\) 717.818 0.123083
\(19\) − 726.021i − 0.105849i −0.998599 − 0.0529247i \(-0.983146\pi\)
0.998599 − 0.0529247i \(-0.0168543\pi\)
\(20\) −1592.75 + 3307.38i −0.199094 + 0.413423i
\(21\) −9841.48 − 1959.67i −1.06268 − 0.211604i
\(22\) −9696.23 + 4669.46i −0.910615 + 0.438529i
\(23\) 8368.08 − 4029.86i 0.687769 − 0.331212i −0.0571514 − 0.998366i \(-0.518202\pi\)
0.744920 + 0.667153i \(0.232487\pi\)
\(24\) 5163.06 + 1178.44i 0.373485 + 0.0852456i
\(25\) −548.557 + 2403.39i −0.0351076 + 0.153817i
\(26\) −3850.61 − 7995.88i −0.219084 − 0.454932i
\(27\) −7642.87 − 15870.6i −0.388298 − 0.806309i
\(28\) −10017.8 − 4485.12i −0.456350 − 0.204315i
\(29\) −28441.0 − 13696.5i −1.16614 − 0.561585i −0.252298 − 0.967650i \(-0.581186\pi\)
−0.913844 + 0.406065i \(0.866901\pi\)
\(30\) −18985.0 −0.703147
\(31\) − 21826.8i − 0.732665i −0.930484 − 0.366332i \(-0.880613\pi\)
0.930484 − 0.366332i \(-0.119387\pi\)
\(32\) 5218.97 + 2513.32i 0.159270 + 0.0767005i
\(33\) −43515.2 − 34702.2i −1.21087 − 0.965640i
\(34\) −12017.4 − 2742.88i −0.305754 − 0.0697864i
\(35\) 38590.0 + 7684.17i 0.900059 + 0.179223i
\(36\) 903.567 + 3958.79i 0.0193666 + 0.0848505i
\(37\) 11206.7 + 5396.86i 0.221245 + 0.106546i 0.541223 − 0.840879i \(-0.317961\pi\)
−0.319979 + 0.947425i \(0.603676\pi\)
\(38\) 4004.03 − 913.893i 0.0729703 − 0.0166550i
\(39\) 28616.8 − 35884.3i 0.482422 − 0.604938i
\(40\) −20245.2 − 4620.84i −0.316331 − 0.0722006i
\(41\) −42261.8 + 33702.7i −0.613192 + 0.489004i −0.880145 − 0.474705i \(-0.842555\pi\)
0.266953 + 0.963710i \(0.413983\pi\)
\(42\) −1580.54 − 56742.8i −0.0213332 − 0.765883i
\(43\) −28339.6 + 35536.8i −0.356442 + 0.446964i −0.927431 − 0.373993i \(-0.877988\pi\)
0.570989 + 0.820957i \(0.306560\pi\)
\(44\) −37957.5 − 47597.2i −0.445594 − 0.558757i
\(45\) −6315.94 − 13115.2i −0.0693107 − 0.143925i
\(46\) 32758.2 + 41077.5i 0.336548 + 0.422018i
\(47\) −55565.4 + 12682.4i −0.535194 + 0.122155i −0.481572 − 0.876407i \(-0.659934\pi\)
−0.0536224 + 0.998561i \(0.517077\pi\)
\(48\) 29957.8i 0.270886i
\(49\) −19754.0 + 115979.i −0.167906 + 0.985803i
\(50\) −13945.2 −0.111562
\(51\) −14185.4 − 62150.4i −0.106938 − 0.468526i
\(52\) 39250.4 − 31301.2i 0.279148 − 0.222613i
\(53\) −52611.9 + 25336.6i −0.353392 + 0.170185i −0.602154 − 0.798380i \(-0.705691\pi\)
0.248762 + 0.968565i \(0.419976\pi\)
\(54\) 77906.1 − 62128.0i 0.494755 − 0.394554i
\(55\) 170630. + 136073.i 1.02558 + 0.817870i
\(56\) 12125.4 − 60894.1i 0.0690451 − 0.346746i
\(57\) 13243.1 + 16606.3i 0.0715096 + 0.0896701i
\(58\) 39735.8 − 174094.i 0.203656 − 0.892276i
\(59\) −274383. − 218813.i −1.33598 − 1.06541i −0.991973 − 0.126446i \(-0.959643\pi\)
−0.344011 − 0.938966i \(-0.611786\pi\)
\(60\) −23897.7 − 104703.i −0.110637 − 0.484734i
\(61\) −31421.0 + 65246.3i −0.138430 + 0.287453i −0.958646 − 0.284601i \(-0.908139\pi\)
0.820216 + 0.572054i \(0.193853\pi\)
\(62\) 120375. − 27474.9i 0.505083 − 0.115282i
\(63\) 38673.2 − 19969.1i 0.154664 − 0.0798614i
\(64\) −7291.57 + 31946.4i −0.0278151 + 0.121866i
\(65\) −112211. + 140708.i −0.408597 + 0.512365i
\(66\) 136608. − 283669.i 0.475165 − 0.986690i
\(67\) 45489.4 0.151247 0.0756233 − 0.997136i \(-0.475905\pi\)
0.0756233 + 0.997136i \(0.475905\pi\)
\(68\) − 69728.7i − 0.221761i
\(69\) −117896. + 244814.i −0.358882 + 0.745227i
\(70\) 6197.54 + 222498.i 0.0180686 + 0.648681i
\(71\) −465423. + 224136.i −1.30039 + 0.626234i −0.950549 − 0.310575i \(-0.899478\pi\)
−0.349840 + 0.936809i \(0.613764\pi\)
\(72\) −20695.4 + 9966.39i −0.0554469 + 0.0267018i
\(73\) −236081. − 53884.0i −0.606866 − 0.138513i −0.0919642 − 0.995762i \(-0.529315\pi\)
−0.514902 + 0.857249i \(0.672172\pi\)
\(74\) −15657.2 + 68598.7i −0.0386384 + 0.169286i
\(75\) −31292.1 − 64978.6i −0.0741738 − 0.154023i
\(76\) 10080.3 + 20931.9i 0.0229632 + 0.0476835i
\(77\) −392494. + 521313.i −0.859727 + 1.14189i
\(78\) 233925. + 112652.i 0.492938 + 0.237386i
\(79\) 365705. 0.741737 0.370869 − 0.928685i \(-0.379060\pi\)
0.370869 + 0.928685i \(0.379060\pi\)
\(80\) − 117469.i − 0.229432i
\(81\) 547649. + 263734.i 1.03050 + 0.496262i
\(82\) −239069. − 190651.i −0.433592 − 0.345778i
\(83\) 548050. + 125089.i 0.958486 + 0.218768i 0.673023 − 0.739621i \(-0.264995\pi\)
0.285463 + 0.958390i \(0.407853\pi\)
\(84\) 310948. − 80142.7i 0.524627 − 0.135215i
\(85\) 55623.3 + 243702.i 0.0905733 + 0.396828i
\(86\) −231659. − 111561.i −0.364212 − 0.175395i
\(87\) 900364. − 205502.i 1.36729 − 0.312075i
\(88\) 214720. − 269250.i 0.315082 − 0.395101i
\(89\) 975174. + 222577.i 1.38329 + 0.315726i 0.848471 − 0.529243i \(-0.177524\pi\)
0.534816 + 0.844969i \(0.320381\pi\)
\(90\) 64380.2 − 51341.5i 0.0883131 − 0.0704273i
\(91\) −429894. − 323665.i −0.570476 − 0.429509i
\(92\) −185309. + 232370.i −0.237975 + 0.298412i
\(93\) 398134. + 499245.i 0.494972 + 0.620676i
\(94\) −139888. − 290481.i −0.168421 − 0.349730i
\(95\) −51928.2 − 65115.9i −0.0605665 − 0.0759480i
\(96\) −165218. + 37709.9i −0.186743 + 0.0426228i
\(97\) 1.44607e6i 1.58443i 0.610242 + 0.792215i \(0.291072\pi\)
−0.610242 + 0.792215i \(0.708928\pi\)
\(98\) −664491. + 37046.8i −0.706010 + 0.0393615i
\(99\) 241411. 0.248801
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 98.7.f.a.13.17 ✓ 168
49.34 odd 14 inner 98.7.f.a.83.17 yes 168
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
98.7.f.a.13.17 ✓ 168 1.1 even 1 trivial
98.7.f.a.83.17 yes 168 49.34 odd 14 inner