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Properties

Label 98.7.f.a.13.11
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 13.11
Character \(\chi\) \(=\) 98.13
Dual form 98.7.f.a.83.11

$q$-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} +(22.5591 - 17.9903i) q^{3} +(-28.8310 + 13.8843i) q^{4} +(29.8758 - 23.8251i) q^{5} +(-127.614 - 101.768i) q^{6} +(117.501 + 322.246i) q^{7} +(112.864 + 141.527i) q^{8} +(23.0449 - 100.966i) q^{9} +(-169.003 - 134.775i) q^{10} +(453.698 + 1987.78i) q^{11} +(-400.619 + 831.895i) q^{12} +(2809.53 - 641.256i) q^{13} +(1629.29 - 1053.65i) q^{14} +(245.349 - 1074.95i) q^{15} +(638.454 - 800.595i) q^{16} +(-2948.65 + 6122.93i) q^{17} -585.840 q^{18} +705.453i q^{19} +(-530.553 + 1101.71i) q^{20} +(8448.01 + 5155.71i) q^{21} +(10391.6 - 5004.31i) q^{22} +(7047.55 - 3393.92i) q^{23} +(5092.21 + 1162.26i) q^{24} +(-3151.97 + 13809.7i) q^{25} +(-7073.08 - 14687.4i) q^{26} +(7830.07 + 16259.3i) q^{27} +(-7861.82 - 7659.27i) q^{28} +(-29732.8 - 14318.6i) q^{29} -6237.19 q^{30} -26296.1i q^{31} +(-5218.97 - 2513.32i) q^{32} +(45995.7 + 36680.4i) q^{33} +(37479.8 + 8554.52i) q^{34} +(11188.0 + 6827.88i) q^{35} +(737.437 + 3230.92i) q^{36} +(-12539.5 - 6038.70i) q^{37} +(3890.59 - 888.002i) q^{38} +(51844.0 - 65010.3i) q^{39} +(6743.78 + 1539.22i) q^{40} +(-1031.05 + 822.237i) q^{41} +(17799.8 - 53080.8i) q^{42} +(64165.9 - 80461.4i) q^{43} +(-40679.5 - 51010.4i) q^{44} +(-1717.05 - 3565.49i) q^{45} +(-27588.8 - 34595.2i) q^{46} +(117626. - 26847.4i) q^{47} -29546.7i q^{48} +(-90036.2 + 75728.2i) q^{49} +80128.2 q^{50} +(43634.4 + 191175. i) q^{51} +(-72098.1 + 57496.3i) q^{52} +(255306. - 122949. i) q^{53} +(79814.2 - 63649.7i) q^{54} +(60913.7 + 48577.0i) q^{55} +(-32344.9 + 52999.4i) q^{56} +(12691.3 + 15914.4i) q^{57} +(-41540.5 + 182001. i) q^{58} +(120760. + 96303.1i) q^{59} +(7851.18 + 34398.3i) q^{60} +(31110.1 - 64600.9i) q^{61} +(-145024. + 33100.7i) q^{62} +(35243.8 - 4437.47i) q^{63} +(-7291.57 + 31946.4i) q^{64} +(68658.7 - 86095.3i) q^{65} +(144395. - 299840. i) q^{66} -404330. q^{67} -217470. i q^{68} +(97928.7 - 203351. i) q^{69} +(23572.9 - 70296.7i) q^{70} +(-388430. + 187058. i) q^{71} +(16890.4 - 8133.96i) q^{72} +(-224071. - 51142.8i) q^{73} +(-17519.3 + 76756.9i) q^{74} +(177334. + 368238. i) q^{75} +(-9794.70 - 20338.9i) q^{76} +(-587245. + 379768. i) q^{77} +(-423793. - 204088. i) q^{78} -840694. q^{79} -39129.6i q^{80} +(537169. + 258687. i) q^{81} +(5832.52 + 4651.28i) q^{82} +(988077. + 225522. i) q^{83} +(-315148. - 31350.0i) q^{84} +(57786.5 + 253179. i) q^{85} +(-524517. - 252594. i) q^{86} +(-928339. + 211887. i) q^{87} +(-230118. + 288559. i) q^{88} +(-375362. - 85673.8i) q^{89} +(-17502.4 + 13957.7i) q^{90} +(536763. + 830011. i) q^{91} +(-156066. + 195700. i) q^{92} +(-473074. - 593216. i) q^{93} +(-296128. - 614916. i) q^{94} +(16807.5 + 21075.9i) q^{95} +(-162951. + 37192.4i) q^{96} +864171. i q^{97} +(530978. + 401228. i) q^{98} +211154. 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
\(3\) 22.5591 − 17.9903i 0.835522 − 0.666307i −0.109257 − 0.994013i \(-0.534847\pi\)
0.944779 + 0.327707i \(0.106276\pi\)
\(4\) −28.8310 + 13.8843i −0.450484 + 0.216942i
\(5\) 29.8758 − 23.8251i 0.239006 − 0.190601i −0.496660 − 0.867945i \(-0.665441\pi\)
0.735666 + 0.677344i \(0.236869\pi\)
\(6\) −127.614 − 101.768i −0.590803 − 0.471150i
\(7\) 117.501 + 322.246i 0.342567 + 0.939493i
\(8\) 112.864 + 141.527i 0.220437 + 0.276419i
\(9\) 23.0449 − 100.966i 0.0316117 − 0.138500i
\(10\) −169.003 − 134.775i −0.169003 − 0.134775i
\(11\) 453.698 + 1987.78i 0.340870 + 1.49345i 0.797242 + 0.603660i \(0.206292\pi\)
−0.456372 + 0.889789i \(0.650851\pi\)
\(12\) −400.619 + 831.895i −0.231840 + 0.481420i
\(13\) 2809.53 − 641.256i 1.27880 − 0.291878i 0.471420 − 0.881909i \(-0.343742\pi\)
0.807381 + 0.590031i \(0.200884\pi\)
\(14\) 1629.29 − 1053.65i 0.593765 − 0.383984i
\(15\) 245.349 − 1074.95i 0.0726961 − 0.318503i
\(16\) 638.454 − 800.595i 0.155872 − 0.195458i
\(17\) −2948.65 + 6122.93i −0.600173 + 1.24627i 0.350642 + 0.936510i \(0.385964\pi\)
−0.950814 + 0.309762i \(0.899751\pi\)
\(18\) −585.840 −0.100453
\(19\) 705.453i 0.102851i 0.998677 + 0.0514253i \(0.0163764\pi\)
−0.998677 + 0.0514253i \(0.983624\pi\)
\(20\) −530.553 + 1101.71i −0.0663192 + 0.137713i
\(21\) 8448.01 + 5155.71i 0.912213 + 0.556712i
\(22\) 10391.6 − 5004.31i 0.975917 − 0.469977i
\(23\) 7047.55 − 3393.92i 0.579235 − 0.278945i −0.121239 − 0.992623i \(-0.538687\pi\)
0.700473 + 0.713679i \(0.252972\pi\)
\(24\) 5092.21 + 1162.26i 0.368360 + 0.0840757i
\(25\) −3151.97 + 13809.7i −0.201726 + 0.883818i
\(26\) −7073.08 − 14687.4i −0.402429 − 0.835652i
\(27\) 7830.07 + 16259.3i 0.397809 + 0.826058i
\(28\) −7861.82 − 7659.27i −0.358137 − 0.348910i
\(29\) −29732.8 − 14318.6i −1.21911 − 0.587091i −0.290044 − 0.957013i \(-0.593670\pi\)
−0.929062 + 0.369923i \(0.879384\pi\)
\(30\) −6237.19 −0.231007
\(31\) − 26296.1i − 0.882685i −0.897339 − 0.441343i \(-0.854502\pi\)
0.897339 − 0.441343i \(-0.145498\pi\)
\(32\) −5218.97 − 2513.32i −0.159270 − 0.0767005i
\(33\) 45995.7 + 36680.4i 1.27990 + 1.02069i
\(34\) 37479.8 + 8554.52i 0.953587 + 0.217650i
\(35\) 11188.0 + 6827.88i 0.260944 + 0.159251i
\(36\) 737.437 + 3230.92i 0.0158058 + 0.0692499i
\(37\) −12539.5 − 6038.70i −0.247557 − 0.119217i 0.305991 − 0.952034i \(-0.401012\pi\)
−0.553548 + 0.832817i \(0.686726\pi\)
\(38\) 3890.59 − 888.002i 0.0709030 − 0.0161831i
\(39\) 51844.0 − 65010.3i 0.873986 − 1.09594i
\(40\) 6743.78 + 1539.22i 0.105372 + 0.0240504i
\(41\) −1031.05 + 822.237i −0.0149599 + 0.0119301i −0.630941 − 0.775831i \(-0.717331\pi\)
0.615981 + 0.787761i \(0.288760\pi\)
\(42\) 17799.8 − 53080.8i 0.240252 − 0.716456i
\(43\) 64165.9 − 80461.4i 0.807047 − 1.01200i −0.192482 − 0.981301i \(-0.561654\pi\)
0.999529 − 0.0307040i \(-0.00977493\pi\)
\(44\) −40679.5 − 51010.4i −0.477548 − 0.598827i
\(45\) −1717.05 − 3565.49i −0.0188428 − 0.0391275i
\(46\) −27588.8 − 34595.2i −0.283439 − 0.355421i
\(47\) 117626. − 26847.4i 1.13295 − 0.258588i 0.385370 − 0.922762i \(-0.374074\pi\)
0.747578 + 0.664174i \(0.231217\pi\)
\(48\) − 29546.7i − 0.267168i
\(49\) −90036.2 + 75728.2i −0.765295 + 0.643679i
\(50\) 80128.2 0.641026
\(51\) 43634.4 + 191175.i 0.328941 + 1.44119i
\(52\) −72098.1 + 57496.3i −0.512759 + 0.408912i
\(53\) 255306. − 122949.i 1.71488 − 0.825842i 0.724206 − 0.689584i \(-0.242207\pi\)
0.990673 − 0.136258i \(-0.0435077\pi\)
\(54\) 79814.2 − 63649.7i 0.506873 − 0.404218i
\(55\) 60913.7 + 48577.0i 0.366123 + 0.291973i
\(56\) −32344.9 + 52999.4i −0.184179 + 0.301791i
\(57\) 12691.3 + 15914.4i 0.0685301 + 0.0859340i
\(58\) −41540.5 + 182001.i −0.212906 + 0.932802i
\(59\) 120760. + 96303.1i 0.587987 + 0.468904i 0.871724 − 0.489998i \(-0.163003\pi\)
−0.283736 + 0.958902i \(0.591574\pi\)
\(60\) 7851.18 + 34398.3i 0.0363481 + 0.159251i
\(61\) 31110.1 − 64600.9i 0.137061 − 0.284609i −0.821129 − 0.570742i \(-0.806656\pi\)
0.958190 + 0.286133i \(0.0923699\pi\)
\(62\) −145024. + 33100.7i −0.608504 + 0.138887i
\(63\) 35243.8 − 4437.47i 0.140949 − 0.0177465i
\(64\) −7291.57 + 31946.4i −0.0278151 + 0.121866i
\(65\) 68658.7 − 86095.3i 0.250009 − 0.313501i
\(66\) 144395. − 299840.i 0.502251 − 1.04294i
\(67\) −404330. −1.34435 −0.672174 − 0.740393i \(-0.734639\pi\)
−0.672174 + 0.740393i \(0.734639\pi\)
\(68\) − 217470.i − 0.691628i
\(69\) 97928.7 − 203351.i 0.298101 − 0.619012i
\(70\) 23572.9 − 70296.7i 0.0687256 − 0.204947i
\(71\) −388430. + 187058.i −1.08527 + 0.522639i −0.888998 − 0.457910i \(-0.848598\pi\)
−0.196273 + 0.980549i \(0.562884\pi\)
\(72\) 16890.4 − 8133.96i 0.0452524 − 0.0217924i
\(73\) −224071. − 51142.8i −0.575993 − 0.131467i −0.0754060 − 0.997153i \(-0.524025\pi\)
−0.500587 + 0.865686i \(0.666882\pi\)
\(74\) −17519.3 + 76756.9i −0.0432335 + 0.189418i
\(75\) 177334. + 368238.i 0.420348 + 0.872861i
\(76\) −9794.70 − 20338.9i −0.0223126 − 0.0463326i
\(77\) −587245. + 379768.i −1.28631 + 0.831852i
\(78\) −423793. − 204088.i −0.893038 − 0.430064i
\(79\) −840694. −1.70513 −0.852564 − 0.522624i \(-0.824953\pi\)
−0.852564 + 0.522624i \(0.824953\pi\)
\(80\) − 39129.6i − 0.0764250i
\(81\) 537169. + 258687.i 1.01078 + 0.486766i
\(82\) 5832.52 + 4651.28i 0.0105783 + 0.00843589i
\(83\) 988077. + 225522.i 1.72805 + 0.394416i 0.967101 − 0.254393i \(-0.0818758\pi\)
0.760951 + 0.648810i \(0.224733\pi\)
\(84\) −315148. − 31350.0i −0.531712 − 0.0528931i
\(85\) 57786.5 + 253179.i 0.0940956 + 0.412260i
\(86\) −524517. − 252594.i −0.824639 − 0.397125i
\(87\) −928339. + 211887.i −1.40977 + 0.321771i
\(88\) −230118. + 288559.i −0.337678 + 0.423434i
\(89\) −375362. − 85673.8i −0.532451 − 0.121528i −0.0521619 − 0.998639i \(-0.516611\pi\)
−0.480289 + 0.877110i \(0.659468\pi\)
\(90\) −17502.4 + 13957.7i −0.0240088 + 0.0191464i
\(91\) 536763. + 830011.i 0.712293 + 1.10144i
\(92\) −156066. + 195700.i −0.200421 + 0.251321i
\(93\) −473074. − 593216.i −0.588139 − 0.737503i
\(94\) −296128. − 614916.i −0.356530 − 0.740342i
\(95\) 16807.5 + 21075.9i 0.0196034 + 0.0245819i
\(96\) −162951. + 37192.4i −0.184180 + 0.0420379i
\(97\) 864171.i 0.946857i 0.880832 + 0.473429i \(0.156984\pi\)
−0.880832 + 0.473429i \(0.843016\pi\)
\(98\) 530978. + 401228.i 0.564155 + 0.426297i
\(99\) 211154. 0.217618
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.13.11 ✓ 168
49.34 odd 14 inner 98.7.f.a.83.11 yes 168
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
98.7.f.a.13.11 ✓ 168 1.1 even 1 trivial
98.7.f.a.83.11 yes 168 49.34 odd 14 inner