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Properties

Label 98.7.f.a.13.12
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 13.12
Character \(\chi\) \(=\) 98.13
Dual form 98.7.f.a.83.12

$q$-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} +(26.3407 - 21.0060i) q^{3} +(-28.8310 + 13.8843i) q^{4} +(149.526 - 119.243i) q^{5} +(-149.006 - 118.828i) q^{6} +(-311.552 + 143.473i) q^{7} +(112.864 + 141.527i) q^{8} +(90.3627 - 395.905i) q^{9} +(-845.848 - 674.541i) q^{10} +(-379.928 - 1664.57i) q^{11} +(-467.776 + 971.346i) q^{12} +(-2611.05 + 595.956i) q^{13} +(1183.43 + 1537.62i) q^{14} +(1433.80 - 6281.90i) q^{15} +(638.454 - 800.595i) q^{16} +(-1386.93 + 2879.98i) q^{17} -2297.17 q^{18} -11552.7i q^{19} +(-2655.39 + 5513.96i) q^{20} +(-5192.70 + 10323.6i) q^{21} +(-8701.92 + 4190.62i) q^{22} +(11338.4 - 5460.30i) q^{23} +(5945.82 + 1357.09i) q^{24} +(4662.26 - 20426.7i) q^{25} +(6573.42 + 13649.9i) q^{26} +(4720.35 + 9801.91i) q^{27} +(6990.33 - 8462.14i) q^{28} +(-25914.0 - 12479.5i) q^{29} -36449.7 q^{30} -24876.5i q^{31} +(-5218.97 - 2513.32i) q^{32} +(-44973.6 - 35865.3i) q^{33} +(17629.0 + 4023.70i) q^{34} +(-29477.0 + 58603.4i) q^{35} +(2891.61 + 12669.0i) q^{36} +(55918.3 + 26928.8i) q^{37} +(-63713.7 + 14542.2i) q^{38} +(-56258.3 + 70545.7i) q^{39} +(33752.2 + 7703.71i) q^{40} +(-205.012 + 163.492i) q^{41} +(63471.6 + 15642.8i) q^{42} +(-77485.9 + 97164.3i) q^{43} +(34065.1 + 42716.3i) q^{44} +(-33697.3 - 69973.3i) q^{45} +(-44386.1 - 55658.5i) q^{46} +(805.507 - 183.852i) q^{47} -34499.6i q^{48} +(76480.0 - 89398.5i) q^{49} -118523. q^{50} +(23964.3 + 104995. i) q^{51} +(67004.9 - 53434.6i) q^{52} +(-205973. + 99191.6i) q^{53} +(48116.0 - 38371.2i) q^{54} +(-255298. - 203593. i) q^{55} +(-55468.1 - 27900.0i) q^{56} +(-242677. - 304307. i) q^{57} +(-36205.2 + 158625. i) q^{58} +(104997. + 83732.1i) q^{59} +(45881.7 + 201021. i) q^{60} +(186419. - 387103. i) q^{61} +(-137195. + 31313.8i) q^{62} +(28649.0 + 136309. i) q^{63} +(-7291.57 + 31946.4i) q^{64} +(-319357. + 400461. i) q^{65} +(-141186. + 293177. i) q^{66} +51345.9 q^{67} -102289. i q^{68} +(183963. - 382003. i) q^{69} +(360304. + 88798.2i) q^{70} +(428359. - 206287. i) q^{71} +(66229.7 - 31894.6i) q^{72} +(-19581.3 - 4469.30i) q^{73} +(78125.0 - 342288. i) q^{74} +(-306277. - 635990. i) q^{75} +(160402. + 333077. i) q^{76} +(357188. + 464091. i) q^{77} +(459878. + 221465. i) q^{78} +609881. q^{79} -195841. i q^{80} +(596957. + 287479. i) q^{81} +(1159.72 + 924.849i) q^{82} +(846421. + 193190. i) q^{83} +(6374.44 - 369738. i) q^{84} +(136036. + 596015. i) q^{85} +(633400. + 305029. i) q^{86} +(-944740. + 215631. i) q^{87} +(192701. - 241640. i) q^{88} +(289160. + 65999.0i) q^{89} +(-343487. + 273922. i) q^{90} +(727975. - 560286. i) q^{91} +(-251086. + 314852. i) q^{92} +(-522557. - 655265. i) q^{93} +(-2027.89 - 4210.97i) q^{94} +(-1.37759e6 - 1.72744e6i) q^{95} +(-190266. + 43427.0i) q^{96} -1.24620e6i q^{97} +(-589306. - 309257. i) q^{98} -693344. 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
\(3\) 26.3407 − 21.0060i 0.975582 − 0.778001i 0.000535184 − 1.00000i \(-0.499830\pi\)
0.975047 + 0.221999i \(0.0712582\pi\)
\(4\) −28.8310 + 13.8843i −0.450484 + 0.216942i
\(5\) 149.526 − 119.243i 1.19621 − 0.953945i 0.196562 − 0.980491i \(-0.437022\pi\)
0.999648 + 0.0265459i \(0.00845082\pi\)
\(6\) −149.006 − 118.828i −0.689841 − 0.550130i
\(7\) −311.552 + 143.473i −0.908314 + 0.418288i
\(8\) 112.864 + 141.527i 0.220437 + 0.276419i
\(9\) 90.3627 − 395.905i 0.123954 − 0.543079i
\(10\) −845.848 − 674.541i −0.845848 − 0.674541i
\(11\) −379.928 − 1664.57i −0.285445 − 1.25062i −0.890702 − 0.454588i \(-0.849787\pi\)
0.605257 − 0.796031i \(-0.293071\pi\)
\(12\) −467.776 + 971.346i −0.270704 + 0.562122i
\(13\) −2611.05 + 595.956i −1.18846 + 0.271259i −0.770641 − 0.637270i \(-0.780064\pi\)
−0.417822 + 0.908529i \(0.637206\pi\)
\(14\) 1183.43 + 1537.62i 0.431279 + 0.560356i
\(15\) 1433.80 − 6281.90i 0.424831 − 1.86130i
\(16\) 638.454 − 800.595i 0.155872 − 0.195458i
\(17\) −1386.93 + 2879.98i −0.282297 + 0.586197i −0.993110 − 0.117185i \(-0.962613\pi\)
0.710813 + 0.703381i \(0.248327\pi\)
\(18\) −2297.17 −0.393891
\(19\) − 11552.7i − 1.68432i −0.539229 − 0.842159i \(-0.681284\pi\)
0.539229 − 0.842159i \(-0.318716\pi\)
\(20\) −2655.39 + 5513.96i −0.331923 + 0.689246i
\(21\) −5192.70 + 10323.6i −0.560706 + 1.11474i
\(22\) −8701.92 + 4190.62i −0.817235 + 0.393560i
\(23\) 11338.4 − 5460.30i 0.931900 − 0.448779i 0.0945952 − 0.995516i \(-0.469844\pi\)
0.837305 + 0.546736i \(0.184130\pi\)
\(24\) 5945.82 + 1357.09i 0.430109 + 0.0981695i
\(25\) 4662.26 − 20426.7i 0.298385 − 1.30731i
\(26\) 6573.42 + 13649.9i 0.374000 + 0.776619i
\(27\) 4720.35 + 9801.91i 0.239819 + 0.497989i
\(28\) 6990.33 − 8462.14i 0.318437 − 0.385484i
\(29\) −25914.0 − 12479.5i −1.06253 − 0.511687i −0.180838 − 0.983513i \(-0.557881\pi\)
−0.881692 + 0.471825i \(0.843595\pi\)
\(30\) −36449.7 −1.34999
\(31\) − 24876.5i − 0.835035i −0.908669 − 0.417517i \(-0.862900\pi\)
0.908669 − 0.417517i \(-0.137100\pi\)
\(32\) −5218.97 − 2513.32i −0.159270 − 0.0767005i
\(33\) −44973.6 − 35865.3i −1.25146 − 0.998004i
\(34\) 17629.0 + 4023.70i 0.448529 + 0.102374i
\(35\) −29477.0 + 58603.4i −0.687510 + 1.36684i
\(36\) 2891.61 + 12669.0i 0.0619771 + 0.271540i
\(37\) 55918.3 + 26928.8i 1.10395 + 0.531633i 0.894898 − 0.446272i \(-0.147248\pi\)
0.209050 + 0.977905i \(0.432963\pi\)
\(38\) −63713.7 + 14542.2i −1.16113 + 0.265021i
\(39\) −56258.3 + 70545.7i −0.948403 + 1.18926i
\(40\) 33752.2 + 7703.71i 0.527378 + 0.120370i
\(41\) −205.012 + 163.492i −0.00297460 + 0.00237216i −0.624976 − 0.780644i \(-0.714891\pi\)
0.622001 + 0.783016i \(0.286320\pi\)
\(42\) 63471.6 + 15642.8i 0.856705 + 0.211138i
\(43\) −77485.9 + 97164.3i −0.974580 + 1.22208i 0.000447526 1.00000i \(0.499858\pi\)
−0.975027 + 0.222085i \(0.928714\pi\)
\(44\) 34065.1 + 42716.3i 0.399900 + 0.501459i
\(45\) −33697.3 − 69973.3i −0.369793 − 0.767882i
\(46\) −44386.1 − 55658.5i −0.456009 − 0.571818i
\(47\) 805.507 − 183.852i 0.00775847 − 0.00177082i −0.218640 − 0.975806i \(-0.570162\pi\)
0.226398 + 0.974035i \(0.427305\pi\)
\(48\) − 34499.6i − 0.311954i
\(49\) 76480.0 − 89398.5i 0.650069 − 0.759875i
\(50\) −118523. −0.948180
\(51\) 23964.3 + 104995.i 0.180657 + 0.791510i
\(52\) 67004.9 − 53434.6i 0.476537 − 0.380025i
\(53\) −205973. + 99191.6i −1.38351 + 0.666265i −0.969746 − 0.244117i \(-0.921502\pi\)
−0.413768 + 0.910382i \(0.635788\pi\)
\(54\) 48116.0 − 38371.2i 0.305568 − 0.243682i
\(55\) −255298. − 203593.i −1.53447 − 1.22370i
\(56\) −55468.1 − 27900.0i −0.315849 − 0.158869i
\(57\) −242677. − 304307.i −1.31040 − 1.64319i
\(58\) −36205.2 + 158625.i −0.185561 + 0.812997i
\(59\) 104997. + 83732.1i 0.511234 + 0.407696i 0.844842 − 0.535016i \(-0.179695\pi\)
−0.333607 + 0.942712i \(0.608266\pi\)
\(60\) 45881.7 + 201021.i 0.212415 + 0.930652i
\(61\) 186419. − 387103.i 0.821297 − 1.70544i 0.119766 − 0.992802i \(-0.461786\pi\)
0.701531 − 0.712639i \(-0.252500\pi\)
\(62\) −137195. + 31313.8i −0.575655 + 0.131389i
\(63\) 28649.0 + 136309.i 0.114574 + 0.545135i
\(64\) −7291.57 + 31946.4i −0.0278151 + 0.121866i
\(65\) −319357. + 400461.i −1.16288 + 1.45821i
\(66\) −141186. + 293177.i −0.491090 + 1.01976i
\(67\) 51345.9 0.170719 0.0853594 − 0.996350i \(-0.472796\pi\)
0.0853594 + 0.996350i \(0.472796\pi\)
\(68\) − 102289.i − 0.325315i
\(69\) 183963. − 382003.i 0.559994 − 1.16284i
\(70\) 360304. + 88798.2i 1.05045 + 0.258887i
\(71\) 428359. − 206287.i 1.19683 − 0.576364i 0.274060 − 0.961713i \(-0.411633\pi\)
0.922771 + 0.385349i \(0.125919\pi\)
\(72\) 66229.7 − 31894.6i 0.177442 − 0.0854514i
\(73\) −19581.3 − 4469.30i −0.0503353 − 0.0114887i 0.197279 − 0.980347i \(-0.436789\pi\)
−0.247614 + 0.968859i \(0.579647\pi\)
\(74\) 78125.0 − 342288.i 0.192795 − 0.844688i
\(75\) −306277. − 635990.i −0.725989 − 1.50753i
\(76\) 160402. + 333077.i 0.365399 + 0.758759i
\(77\) 357188. + 464091.i 0.782393 + 1.01656i
\(78\) 459878. + 221465.i 0.969077 + 0.466683i
\(79\) 609881. 1.23698 0.618492 − 0.785791i \(-0.287744\pi\)
0.618492 + 0.785791i \(0.287744\pi\)
\(80\) − 195841.i − 0.382502i
\(81\) 596957. + 287479.i 1.12328 + 0.540943i
\(82\) 1159.72 + 924.849i 0.00210336 + 0.00167737i
\(83\) 846421. + 193190.i 1.48031 + 0.337871i 0.884986 − 0.465617i \(-0.154168\pi\)
0.595321 + 0.803488i \(0.297025\pi\)
\(84\) 6374.44 − 369738.i 0.0107548 − 0.623815i
\(85\) 136036. + 596015.i 0.221513 + 0.970510i
\(86\) 633400. + 305029.i 0.995825 + 0.479564i
\(87\) −944740. + 215631.i −1.43468 + 0.327456i
\(88\) 192701. − 241640.i 0.282772 − 0.354585i
\(89\) 289160. + 65999.0i 0.410175 + 0.0936197i 0.422629 − 0.906303i \(-0.361107\pi\)
−0.0124545 + 0.999922i \(0.503964\pi\)
\(90\) −343487. + 273922.i −0.471176 + 0.375750i
\(91\) 727975. − 560286.i 0.966033 − 0.743509i
\(92\) −251086. + 314852.i −0.322447 + 0.404336i
\(93\) −522557. − 655265.i −0.649658 − 0.814645i
\(94\) −2027.89 − 4210.97i −0.00244153 − 0.00506989i
\(95\) −1.37759e6 − 1.72744e6i −1.60675 − 2.01480i
\(96\) −190266. + 43427.0i −0.215054 + 0.0490847i
\(97\) − 1.24620e6i − 1.36544i −0.730682 − 0.682718i \(-0.760798\pi\)
0.730682 − 0.682718i \(-0.239202\pi\)
\(98\) −589306. − 309257.i −0.626127 − 0.328580i
\(99\) −693344. −0.714567
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.13.12 ✓ 168
49.34 odd 14 inner 98.7.f.a.83.12 yes 168
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
98.7.f.a.13.12 ✓ 168 1.1 even 1 trivial
98.7.f.a.83.12 yes 168 49.34 odd 14 inner