LMFDB - Embedded newform 208.10.a.l.1.5

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [208,10,Mod(1,208)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(208, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 10, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("208.1"); S:= CuspForms(chi, 10); N := Newforms(S);

Level:	$ N $	$=$	$ 208 = 2^{4} \cdot 13 $
Weight:	$ k $	$=$	$ 10 $
Character orbit:	$[\chi]$	$=$	208.a (trivial)

Newform invariants

comment:select newform

sage:traces = [7,0,-13] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$107.127453922$
Analytic rank:	$1$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - x^{6} - 112810 x^{5} + 1645934 x^{4} + 3493976849 x^{3} - 83049726457 x^{2} + \cdots + 864293655586764 $ x^7 - x^6 - 112810x^5 + 1645934x^4 + 3493976849x^3 - 83049726457x^2 - 21841140807080*x + 864293655586764
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{19}\cdot 3 $
Twist minimal:	no (minimal twist has level 104)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$71.8667$ of defining polynomial
Character	$\chi$	$=$	208.1

$q$-expansion

comment:q-expansion

sage:f.q_expansion() # note that sage often uses an isomorphic number field

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+69.8667 q^{3} -170.421 q^{5} +12012.7 q^{7} -14801.7 q^{9} +36255.5 q^{11} -28561.0 q^{13} -11906.7 q^{15} -523020. q^{17} -706794. q^{19} +839290. q^{21} +26699.1 q^{23} -1.92408e6 q^{25} -2.40933e6 q^{27} -2.38415e6 q^{29} +6.10212e6 q^{31} +2.53305e6 q^{33} -2.04722e6 q^{35} +3.21358e6 q^{37} -1.99546e6 q^{39} -2.58454e7 q^{41} -1.94597e7 q^{43} +2.52251e6 q^{45} +9.09217e6 q^{47} +1.03952e8 q^{49} -3.65417e7 q^{51} +7.10014e7 q^{53} -6.17869e6 q^{55} -4.93814e7 q^{57} +3.54719e7 q^{59} -8.93209e7 q^{61} -1.77808e8 q^{63} +4.86739e6 q^{65} -1.09614e8 q^{67} +1.86538e6 q^{69} -3.14783e8 q^{71} -3.00502e8 q^{73} -1.34429e8 q^{75} +4.35528e8 q^{77} +1.19919e8 q^{79} +1.23009e8 q^{81} -4.28505e8 q^{83} +8.91336e7 q^{85} -1.66572e8 q^{87} +2.06268e8 q^{89} -3.43096e8 q^{91} +4.26335e8 q^{93} +1.20453e8 q^{95} -4.90604e8 q^{97} -5.36641e8 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q - 13 q^{3} - 801 q^{5} + 1091 q^{7} + 87864 q^{9} - 32218 q^{11} - 199927 q^{13} - 58323 q^{15} - 870531 q^{17} + 128950 q^{19} - 719663 q^{21} - 2198844 q^{23} + 992010 q^{25} - 5441383 q^{27} + 6327710 q^{29}+ \cdots - 5204667600 q^{99}+O(q^{100}) $ 7 * q - 13 * q^3 - 801 * q^5 + 1091 * q^7 + 87864 * q^9 - 32218 * q^11 - 199927 * q^13 - 58323 * q^15 - 870531 * q^17 + 128950 * q^19 - 719663 * q^21 - 2198844 * q^23 + 992010 * q^25 - 5441383 * q^27 + 6327710 * q^29 + 1660376 * q^31 + 14199598 * q^33 + 19231769 * q^35 + 5058787 * q^37 + 371293 * q^39 + 14779748 * q^41 - 9405075 * q^43 + 56644022 * q^45 - 133415721 * q^47 + 44544988 * q^49 - 132066169 * q^51 + 43962424 * q^53 - 191344746 * q^55 + 138963078 * q^57 - 22370274 * q^59 + 23851748 * q^61 - 780720706 * q^63 + 22877361 * q^65 - 12058786 * q^67 + 1036453420 * q^69 - 864303791 * q^71 + 670491202 * q^73 - 693324044 * q^75 + 841052238 * q^77 - 1019547572 * q^79 + 900194079 * q^81 - 1345041700 * q^83 + 824066375 * q^85 - 2589172074 * q^87 + 1409158546 * q^89 - 31160051 * q^91 - 445200892 * q^93 - 3285213058 * q^95 + 320206906 * q^97 - 5204667600 * q^99

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	69.8667	0.497994	0.248997	−	0.968504i	$-0.419899\pi$
$3$	69.8667	0.497994	0.248997	+	0.968504i	$0.419899\pi$
$4$	0	0
$5$	−170.421	−0.121943	−0.0609716	−	0.998140i	$-0.519420\pi$
$5$	−170.421	−0.121943	−0.0609716	+	0.998140i	$0.519420\pi$
$6$	0	0
$7$	12012.7	1.89104	0.945520	−	0.325564i	$-0.105554\pi$
$7$	12012.7	1.89104	0.945520	+	0.325564i	$0.105554\pi$
$8$	0	0
$9$	−14801.7	−0.752002
$10$	0	0
$11$	36255.5	0.746632	0.373316	−	0.927704i	$-0.378221\pi$
$11$	36255.5	0.746632	0.373316	+	0.927704i	$0.378221\pi$
$12$	0	0
$13$	−28561.0	−0.277350
$13$	−28561.0	−0.277350
$14$	0	0
$15$	−11906.7	−0.0607270
$16$	0	0
$17$	−523020.	−1.51879	−0.759396	−	0.650628i	$-0.774506\pi$
$17$	−523020.	−1.51879	−0.759396	+	0.650628i	$0.774506\pi$
$18$	0	0
$19$	−706794.	−1.24423	−0.622117	−	0.782924i	$-0.713727\pi$
$19$	−706794.	−1.24423	−0.622117	+	0.782924i	$0.713727\pi$
$20$	0	0
$21$	839290.	0.941727
$22$	0	0
$23$	26699.1	0.0198940	0.00994698	−	0.999951i	$-0.496834\pi$
$23$	26699.1	0.0198940	0.00994698	+	0.999951i	$0.496834\pi$
$24$	0	0
$25$	−1.92408e6	−0.985130
$26$	0	0
$27$	−2.40933e6	−0.872487
$28$	0	0
$29$	−2.38415e6	−0.625953	−0.312977	−	0.949761i	$-0.601326\pi$
$29$	−2.38415e6	−0.625953	−0.312977	+	0.949761i	$0.601326\pi$
$30$	0	0
$31$	6.10212e6	1.18673	0.593367	−	0.804932i	$-0.297798\pi$
$31$	6.10212e6	1.18673	0.593367	+	0.804932i	$0.297798\pi$
$32$	0	0
$33$	2.53305e6	0.371818
$34$	0	0
$35$	−2.04722e6	−0.230600
$36$	0	0
$37$	3.21358e6	0.281891	0.140946	−	0.990017i	$-0.454986\pi$
$37$	3.21358e6	0.281891	0.140946	+	0.990017i	$0.454986\pi$
$38$	0	0
$39$	−1.99546e6	−0.138119
$40$	0	0
$41$	−2.58454e7	−1.42842	−0.714209	−	0.699933i	$-0.753213\pi$
$41$	−2.58454e7	−1.42842	−0.714209	+	0.699933i	$0.753213\pi$
$42$	0	0
$43$	−1.94597e7	−0.868018	−0.434009	−	0.900909i	$-0.642901\pi$
$43$	−1.94597e7	−0.868018	−0.434009	+	0.900909i	$0.642901\pi$
$44$	0	0
$45$	2.52251e6	0.0917015
$46$	0	0
$47$	9.09217e6	0.271786	0.135893	−	0.990724i	$-0.456610\pi$
$47$	9.09217e6	0.271786	0.135893	+	0.990724i	$0.456610\pi$
$48$	0	0
$49$	1.03952e8	2.57603
$50$	0	0
$51$	−3.65417e7	−0.756350
$52$	0	0
$53$	7.10014e7	1.23602	0.618010	−	0.786170i	$-0.287939\pi$
$53$	7.10014e7	1.23602	0.618010	+	0.786170i	$0.287939\pi$
$54$	0	0
$55$	−6.17869e6	−0.0910467
$56$	0	0
$57$	−4.93814e7	−0.619621
$58$	0	0
$59$	3.54719e7	0.381110	0.190555	−	0.981677i	$-0.438971\pi$
$59$	3.54719e7	0.381110	0.190555	+	0.981677i	$0.438971\pi$
$60$	0	0
$61$	−8.93209e7	−0.825978	−0.412989	−	0.910736i	$-0.635515\pi$
$61$	−8.93209e7	−0.825978	−0.412989	+	0.910736i	$0.635515\pi$
$62$	0	0
$63$	−1.77808e8	−1.42207
$64$	0	0
$65$	4.86739e6	0.0338210
$66$	0	0
$67$	−1.09614e8	−0.664553	−0.332276	−	0.943182i	$-0.607817\pi$
$67$	−1.09614e8	−0.664553	−0.332276	+	0.943182i	$0.607817\pi$
$68$	0	0
$69$	1.86538e6	0.00990707
$70$	0	0
$71$	−3.14783e8	−1.47010	−0.735052	−	0.678011i	$-0.762842\pi$
$71$	−3.14783e8	−1.47010	−0.735052	+	0.678011i	$0.762842\pi$
$72$	0	0
$73$	−3.00502e8	−1.23850	−0.619248	−	0.785196i	$-0.712562\pi$
$73$	−3.00502e8	−1.23850	−0.619248	+	0.785196i	$0.712562\pi$
$74$	0	0
$75$	−1.34429e8	−0.490589
$76$	0	0
$77$	4.35528e8	1.41191
$78$	0	0
$79$	1.19919e8	0.346391	0.173195	−	0.984887i	$-0.444591\pi$
$79$	1.19919e8	0.346391	0.173195	+	0.984887i	$0.444591\pi$
$80$	0	0
$81$	1.23009e8	0.317508
$82$	0	0
$83$	−4.28505e8	−0.991071	−0.495535	−	0.868588i	$-0.665028\pi$
$83$	−4.28505e8	−0.991071	−0.495535	+	0.868588i	$0.665028\pi$
$84$	0	0
$85$	8.91336e7	0.185206
$86$	0	0
$87$	−1.66572e8	−0.311721
$88$	0	0
$89$	2.06268e8	0.348479	0.174239	−	0.984703i	$-0.444253\pi$
$89$	2.06268e8	0.348479	0.174239	+	0.984703i	$0.444253\pi$
$90$	0	0
$91$	−3.43096e8	−0.524480
$92$	0	0
$93$	4.26335e8	0.590986
$94$	0	0
$95$	1.20453e8	0.151726
$96$	0	0
$97$	−4.90604e8	−0.562675	−0.281338	−	0.959609i	$-0.590778\pi$
$97$	−4.90604e8	−0.562675	−0.281338	+	0.959609i	$0.590778\pi$
$98$	0	0
$99$	−5.36641e8	−0.561469
$100$	0	0
$101$	1.17120e9	1.11992	0.559959	−	0.828520i	$-0.310817\pi$
$101$	1.17120e9	1.11992	0.559959	+	0.828520i	$0.310817\pi$
$102$	0	0
$103$	1.53711e9	1.34567	0.672835	−	0.739793i	$-0.265077\pi$
$103$	1.53711e9	1.34567	0.672835	+	0.739793i	$0.265077\pi$
$104$	0	0
$105$	−1.43032e8	−0.114837
$106$	0	0
$107$	5.35059e8	0.394616	0.197308	−	0.980342i	$-0.436780\pi$
$107$	5.35059e8	0.394616	0.197308	+	0.980342i	$0.436780\pi$
$108$	0	0
$109$	2.09611e9	1.42231	0.711155	−	0.703036i	$-0.248173\pi$
$109$	2.09611e9	1.42231	0.711155	+	0.703036i	$0.248173\pi$
$110$	0	0
$111$	2.24522e8	0.140380
$112$	0	0
$113$	−1.47697e9	−0.852157	−0.426079	−	0.904686i	$-0.640105\pi$
$113$	−1.47697e9	−0.852157	−0.426079	+	0.904686i	$0.640105\pi$
$114$	0	0
$115$	−4.55008e6	−0.00242593
$116$	0	0
$117$	4.22750e8	0.208568
$118$	0	0
$119$	−6.28291e9	−2.87210
$120$	0	0
$121$	−1.04349e9	−0.442541
$122$	0	0
$123$	−1.80573e9	−0.711344
$124$	0	0
$125$	6.60757e8	0.242073
$126$	0	0
$127$	−1.53809e9	−0.524645	−0.262323	−	0.964980i	$-0.584488\pi$
$127$	−1.53809e9	−0.524645	−0.262323	+	0.964980i	$0.584488\pi$
$128$	0	0
$129$	−1.35959e9	−0.432268
$130$	0	0
$131$	1.59728e9	0.473872	0.236936	−	0.971525i	$-0.423857\pi$
$131$	1.59728e9	0.473872	0.236936	+	0.971525i	$0.423857\pi$
$132$	0	0
$133$	−8.49054e9	−2.35290
$134$	0	0
$135$	4.10600e8	0.106394
$136$	0	0
$137$	−7.12908e9	−1.72898	−0.864492	−	0.502647i	$-0.832360\pi$
$137$	−7.12908e9	−1.72898	−0.864492	+	0.502647i	$0.832360\pi$
$138$	0	0
$139$	−6.73230e9	−1.52967	−0.764833	−	0.644228i	$-0.777179\pi$
$139$	−6.73230e9	−1.52967	−0.764833	+	0.644228i	$0.777179\pi$
$140$	0	0
$141$	6.35240e8	0.135348
$142$	0	0
$143$	−1.03549e9	−0.207078
$144$	0	0
$145$	4.06308e8	0.0763307
$146$	0	0
$147$	7.26279e9	1.28285
$148$	0	0
$149$	−1.14876e10	−1.90937	−0.954686	−	0.297616i	$-0.903808\pi$
$149$	−1.14876e10	−1.90937	−0.954686	+	0.297616i	$0.903808\pi$
$150$	0	0
$151$	7.91167e9	1.23843	0.619216	−	0.785221i	$-0.287451\pi$
$151$	7.91167e9	1.23843	0.619216	+	0.785221i	$0.287451\pi$
$152$	0	0
$153$	7.74156e9	1.14213
$154$	0	0
$155$	−1.03993e9	−0.144714
$156$	0	0
$157$	−1.13340e10	−1.48879	−0.744396	−	0.667739i	$-0.767262\pi$
$157$	−1.13340e10	−1.48879	−0.744396	+	0.667739i	$0.767262\pi$
$158$	0	0
$159$	4.96063e9	0.615531
$160$	0	0
$161$	3.20729e8	0.0376203
$162$	0	0
$163$	1.31887e10	1.46338	0.731692	−	0.681635i	$-0.238731\pi$
$163$	1.31887e10	1.46338	0.731692	+	0.681635i	$0.238731\pi$
$164$	0	0
$165$	−4.31684e8	−0.0453407
$166$	0	0
$167$	−1.55218e10	−1.54425	−0.772127	−	0.635468i	$-0.780807\pi$
$167$	−1.55218e10	−1.54425	−0.772127	+	0.635468i	$0.780807\pi$
$168$	0	0
$169$	8.15731e8	0.0769231
$170$	0	0
$171$	1.04617e10	0.935666
$172$	0	0
$173$	−8.60185e9	−0.730104	−0.365052	−	0.930987i	$-0.618949\pi$
$173$	−8.60185e9	−0.730104	−0.365052	+	0.930987i	$0.618949\pi$
$174$	0	0
$175$	−2.31135e10	−1.86292
$176$	0	0
$177$	2.47830e9	0.189790
$178$	0	0
$179$	−1.40539e10	−1.02320	−0.511598	−	0.859225i	$-0.670946\pi$
$179$	−1.40539e10	−1.02320	−0.511598	+	0.859225i	$0.670946\pi$
$180$	0	0
$181$	−4.83176e9	−0.334620	−0.167310	−	0.985904i	$-0.553508\pi$
$181$	−4.83176e9	−0.334620	−0.167310	+	0.985904i	$0.553508\pi$
$182$	0	0
$183$	−6.24055e9	−0.411332
$184$	0	0
$185$	−5.47661e8	−0.0343747
$186$	0	0
$187$	−1.89624e10	−1.13398
$188$	0	0
$189$	−2.89426e10	−1.64991
$190$	0	0
$191$	3.32432e10	1.80739	0.903696	−	0.428174i	$-0.140843\pi$
$191$	3.32432e10	1.80739	0.903696	+	0.428174i	$0.140843\pi$
$192$	0	0
$193$	−4.50395e9	−0.233660	−0.116830	−	0.993152i	$-0.537273\pi$
$193$	−4.50395e9	−0.233660	−0.116830	+	0.993152i	$0.537273\pi$
$194$	0	0
$195$	3.40068e8	0.0168426
$196$	0	0
$197$	−2.15891e10	−1.02126	−0.510630	−	0.859801i	$-0.670588\pi$
$197$	−2.15891e10	−1.02126	−0.510630	+	0.859801i	$0.670588\pi$
$198$	0	0
$199$	2.46171e7	0.00111275	0.000556376	−	1.00000i	$-0.499823\pi$
$199$	2.46171e7	0.00111275	0.000556376		1.00000i	$0.499823\pi$
$200$	0	0
$201$	−7.65836e9	−0.330943
$202$	0	0
$203$	−2.86401e10	−1.18370
$204$	0	0
$205$	4.40459e9	0.174186
$206$	0	0
$207$	−3.95191e8	−0.0149603
$208$	0	0
$209$	−2.56252e10	−0.928985
$210$	0	0
$211$	−7.09907e9	−0.246565	−0.123282	−	0.992372i	$-0.539342\pi$
$211$	−7.09907e9	−0.246565	−0.123282	+	0.992372i	$0.539342\pi$
$212$	0	0
$213$	−2.19928e10	−0.732104
$214$	0	0
$215$	3.31634e9	0.105849
$216$	0	0
$217$	7.33032e10	2.24416
$218$	0	0
$219$	−2.09951e10	−0.616764
$220$	0	0
$221$	1.49380e10	0.421237
$222$	0	0
$223$	−4.59128e10	−1.24326	−0.621629	−	0.783311i	$-0.713529\pi$
$223$	−4.59128e10	−1.24326	−0.621629	+	0.783311i	$0.713529\pi$
$224$	0	0
$225$	2.84796e10	0.740819
$226$	0	0
$227$	9.67572e7	0.00241861	0.00120931	−	0.999999i	$-0.499615\pi$
$227$	9.67572e7	0.00241861	0.00120931	+	0.999999i	$0.499615\pi$
$228$	0	0
$229$	4.53850e10	1.09057	0.545284	−	0.838252i	$-0.316422\pi$
$229$	4.53850e10	1.09057	0.545284	+	0.838252i	$0.316422\pi$
$230$	0	0
$231$	3.04289e10	0.703124
$232$	0	0
$233$	−1.30315e10	−0.289664	−0.144832	−	0.989456i	$-0.546264\pi$
$233$	−1.30315e10	−0.289664	−0.144832	+	0.989456i	$0.546264\pi$
$234$	0	0
$235$	−1.54950e9	−0.0331425
$236$	0	0
$237$	8.37834e9	0.172501
$238$	0	0
$239$	−8.99182e9	−0.178261	−0.0891306	−	0.996020i	$-0.528409\pi$
$239$	−8.99182e9	−0.178261	−0.0891306	+	0.996020i	$0.528409\pi$
$240$	0	0
$241$	7.97646e10	1.52312	0.761559	−	0.648095i	$-0.224434\pi$
$241$	7.97646e10	1.52312	0.761559	+	0.648095i	$0.224434\pi$
$242$	0	0
$243$	5.60170e10	1.03060
$244$	0	0
$245$	−1.77156e10	−0.314130
$246$	0	0
$247$	2.01868e10	0.345088
$248$	0	0
$249$	−2.99382e10	−0.493548
$250$	0	0
$251$	−7.05270e10	−1.12156	−0.560781	−	0.827964i	$-0.689499\pi$
$251$	−7.05270e10	−1.12156	−0.560781	+	0.827964i	$0.689499\pi$
$252$	0	0
$253$	9.67988e8	0.0148535
$254$	0	0
$255$	6.22746e9	0.0922318
$256$	0	0
$257$	−3.62085e10	−0.517740	−0.258870	−	0.965912i	$-0.583350\pi$
$257$	−3.62085e10	−0.517740	−0.258870	+	0.965912i	$0.583350\pi$
$258$	0	0
$259$	3.86039e10	0.533067
$260$	0	0
$261$	3.52893e10	0.470718
$262$	0	0
$263$	−1.62926e10	−0.209986	−0.104993	−	0.994473i	$-0.533482\pi$
$263$	−1.62926e10	−0.209986	−0.104993	+	0.994473i	$0.533482\pi$
$264$	0	0
$265$	−1.21001e10	−0.150724
$266$	0	0
$267$	1.44112e10	0.173540
$268$	0	0
$269$	8.86274e10	1.03201	0.516003	−	0.856587i	$-0.327419\pi$
$269$	8.86274e10	1.03201	0.516003	+	0.856587i	$0.327419\pi$
$270$	0	0
$271$	6.36764e10	0.717161	0.358581	−	0.933499i	$-0.383261\pi$
$271$	6.36764e10	0.717161	0.358581	+	0.933499i	$0.383261\pi$
$272$	0	0
$273$	−2.39710e10	−0.261188
$274$	0	0
$275$	−6.97585e10	−0.735529
$276$	0	0
$277$	1.19356e11	1.21811	0.609054	−	0.793129i	$-0.291549\pi$
$277$	1.19356e11	1.21811	0.609054	+	0.793129i	$0.291549\pi$
$278$	0	0
$279$	−9.03215e10	−0.892426
$280$	0	0
$281$	−5.66740e10	−0.542257	−0.271129	−	0.962543i	$-0.587397\pi$
$281$	−5.66740e10	−0.542257	−0.271129	+	0.962543i	$0.587397\pi$
$282$	0	0
$283$	1.48510e11	1.37631	0.688155	−	0.725564i	$-0.258421\pi$
$283$	1.48510e11	1.37631	0.688155	+	0.725564i	$0.258421\pi$
$284$	0	0
$285$	8.41561e9	0.0755586
$286$	0	0
$287$	−3.10474e11	−2.70120
$288$	0	0
$289$	1.54962e11	1.30673
$290$	0	0
$291$	−3.42768e10	−0.280209
$292$	0	0
$293$	1.98296e11	1.57185	0.785924	−	0.618323i	$-0.212188\pi$
$293$	1.98296e11	1.57185	0.785924	+	0.618323i	$0.212188\pi$
$294$	0	0
$295$	−6.04515e9	−0.0464738
$296$	0	0
$297$	−8.73513e10	−0.651427
$298$	0	0
$299$	−7.62553e8	−0.00551759
$300$	0	0
$301$	−2.33765e11	−1.64146
$302$	0	0
$303$	8.18281e10	0.557713
$304$	0	0
$305$	1.52221e10	0.100722
$306$	0	0
$307$	2.16742e11	1.39258	0.696289	−	0.717761i	$-0.254833\pi$
$307$	2.16742e11	1.39258	0.696289	+	0.717761i	$0.254833\pi$
$308$	0	0
$309$	1.07393e11	0.670135
$310$	0	0
$311$	2.50503e11	1.51842	0.759210	−	0.650846i	$-0.225586\pi$
$311$	2.50503e11	1.51842	0.759210	+	0.650846i	$0.225586\pi$
$312$	0	0
$313$	−2.73413e11	−1.61017	−0.805083	−	0.593163i	$-0.797879\pi$
$313$	−2.73413e11	−1.61017	−0.805083	+	0.593163i	$0.797879\pi$
$314$	0	0
$315$	3.03022e10	0.173411
$316$	0	0
$317$	−1.39923e11	−0.778258	−0.389129	−	0.921183i	$-0.627224\pi$
$317$	−1.39923e11	−0.778258	−0.389129	+	0.921183i	$0.627224\pi$
$318$	0	0
$319$	−8.64383e10	−0.467357
$320$	0	0
$321$	3.73828e10	0.196516
$322$	0	0
$323$	3.69668e11	1.88973
$324$	0	0
$325$	5.49537e10	0.273226
$326$	0	0
$327$	1.46448e11	0.708302
$328$	0	0
$329$	1.09222e11	0.513959
$330$	0	0
$331$	−3.11665e11	−1.42712	−0.713562	−	0.700592i	$-0.752919\pi$
$331$	−3.11665e11	−1.42712	−0.713562	+	0.700592i	$0.752919\pi$
$332$	0	0
$333$	−4.75663e10	−0.211983
$334$	0	0
$335$	1.86805e10	0.0810377
$336$	0	0
$337$	1.02851e11	0.434386	0.217193	−	0.976129i	$-0.430310\pi$
$337$	1.02851e11	0.434386	0.217193	+	0.976129i	$0.430310\pi$
$338$	0	0
$339$	−1.03191e11	−0.424370
$340$	0	0
$341$	2.21235e11	0.886053
$342$	0	0
$343$	7.63993e11	2.98034
$344$	0	0
$345$	−3.17899e8	−0.00120810
$346$	0	0
$347$	4.31461e11	1.59757	0.798783	−	0.601620i	$-0.205478\pi$
$347$	4.31461e11	1.59757	0.798783	+	0.601620i	$0.205478\pi$
$348$	0	0
$349$	−4.29206e10	−0.154864	−0.0774321	−	0.996998i	$-0.524672\pi$
$349$	−4.29206e10	−0.154864	−0.0774321	+	0.996998i	$0.524672\pi$
$350$	0	0
$351$	6.88128e10	0.241984
$352$	0	0
$353$	5.45378e10	0.186944	0.0934719	−	0.995622i	$-0.470203\pi$
$353$	5.45378e10	0.186944	0.0934719	+	0.995622i	$0.470203\pi$
$354$	0	0
$355$	5.36455e10	0.179269
$356$	0	0
$357$	−4.38966e11	−1.43029
$358$	0	0
$359$	−3.00497e11	−0.954807	−0.477404	−	0.878684i	$-0.658422\pi$
$359$	−3.00497e11	−0.954807	−0.477404	+	0.878684i	$0.658422\pi$
$360$	0	0
$361$	1.76871e11	0.548117
$362$	0	0
$363$	−7.29050e10	−0.220383
$364$	0	0
$365$	5.12118e10	0.151026
$366$	0	0
$367$	−3.68222e11	−1.05953	−0.529764	−	0.848145i	$-0.677719\pi$
$367$	−3.68222e11	−1.05953	−0.529764	+	0.848145i	$0.677719\pi$
$368$	0	0
$369$	3.82554e11	1.07417
$370$	0	0
$371$	8.52921e11	2.33736
$372$	0	0
$373$	−2.21144e11	−0.591543	−0.295771	−	0.955259i	$-0.595577\pi$
$373$	−2.21144e11	−0.591543	−0.295771	+	0.955259i	$0.595577\pi$
$374$	0	0
$375$	4.61649e10	0.120551
$376$	0	0
$377$	6.80936e10	0.173608
$378$	0	0
$379$	−3.34937e11	−0.833847	−0.416924	−	0.908942i	$-0.636892\pi$
$379$	−3.34937e11	−0.833847	−0.416924	+	0.908942i	$0.636892\pi$
$380$	0	0
$381$	−1.07461e11	−0.261270
$382$	0	0
$383$	4.98211e10	0.118309	0.0591546	−	0.998249i	$-0.481159\pi$
$383$	4.98211e10	0.118309	0.0591546	+	0.998249i	$0.481159\pi$
$384$	0	0
$385$	−7.42230e10	−0.172173
$386$	0	0
$387$	2.88036e11	0.652751
$388$	0	0
$389$	−1.47884e11	−0.327452	−0.163726	−	0.986506i	$-0.552351\pi$
$389$	−1.47884e11	−0.327452	−0.163726	+	0.986506i	$0.552351\pi$
$390$	0	0
$391$	−1.39642e10	−0.0302148
$392$	0	0
$393$	1.11597e11	0.235985
$394$	0	0
$395$	−2.04367e10	−0.0422400
$396$	0	0
$397$	1.08303e11	0.218818	0.109409	−	0.993997i	$-0.465104\pi$
$397$	1.08303e11	0.218818	0.109409	+	0.993997i	$0.465104\pi$
$398$	0	0
$399$	−5.93205e11	−1.17173
$400$	0	0
$401$	−3.59927e11	−0.695127	−0.347564	−	0.937656i	$-0.612991\pi$
$401$	−3.59927e11	−0.695127	−0.347564	+	0.937656i	$0.612991\pi$
$402$	0	0
$403$	−1.74283e11	−0.329141
$404$	0	0
$405$	−2.09633e10	−0.0387180
$406$	0	0
$407$	1.16510e11	0.210469
$408$	0	0
$409$	−5.97247e10	−0.105536	−0.0527678	−	0.998607i	$-0.516804\pi$
$409$	−5.97247e10	−0.105536	−0.0527678	+	0.998607i	$0.516804\pi$
$410$	0	0
$411$	−4.98085e11	−0.861024
$412$	0	0
$413$	4.26114e11	0.720694
$414$	0	0
$415$	7.30262e10	0.120854
$416$	0	0
$417$	−4.70363e11	−0.761765
$418$	0	0
$419$	1.00231e12	1.58868	0.794341	−	0.607472i	$-0.207816\pi$
$419$	1.00231e12	1.58868	0.794341	+	0.607472i	$0.207816\pi$
$420$	0	0
$421$	5.14253e11	0.797825	0.398912	−	0.916989i	$-0.369388\pi$
$421$	5.14253e11	0.797825	0.398912	+	0.916989i	$0.369388\pi$
$422$	0	0
$423$	−1.34579e11	−0.204384
$424$	0	0
$425$	1.00633e12	1.49621
$426$	0	0
$427$	−1.07299e12	−1.56196
$428$	0	0
$429$	−7.23464e10	−0.103124
$430$	0	0
$431$	−3.67084e11	−0.512411	−0.256205	−	0.966622i	$-0.582472\pi$
$431$	−3.67084e11	−0.512411	−0.256205	+	0.966622i	$0.582472\pi$
$432$	0	0
$433$	7.24529e11	0.990513	0.495257	−	0.868747i	$-0.335074\pi$
$433$	7.24529e11	0.990513	0.495257	+	0.868747i	$0.335074\pi$
$434$	0	0
$435$	2.83874e10	0.0380123
$436$	0	0
$437$	−1.88708e10	−0.0247527
$438$	0	0
$439$	−9.66365e11	−1.24180	−0.620899	−	0.783891i	$-0.713232\pi$
$439$	−9.66365e11	−1.24180	−0.620899	+	0.783891i	$0.713232\pi$
$440$	0	0
$441$	−1.53866e12	−1.93718
$442$	0	0
$443$	2.90663e10	0.0358569	0.0179285	−	0.999839i	$-0.494293\pi$
$443$	2.90663e10	0.0358569	0.0179285	+	0.999839i	$0.494293\pi$
$444$	0	0
$445$	−3.51523e10	−0.0424946
$446$	0	0
$447$	−8.02599e11	−0.950856
$448$	0	0
$449$	−9.14242e11	−1.06158	−0.530790	−	0.847503i	$-0.678105\pi$
$449$	−9.14242e11	−1.06158	−0.530790	+	0.847503i	$0.678105\pi$
$450$	0	0
$451$	−9.37036e11	−1.06650
$452$	0	0
$453$	5.52762e11	0.616732
$454$	0	0
$455$	5.84707e10	0.0639568
$456$	0	0
$457$	1.16186e12	1.24604	0.623021	−	0.782205i	$-0.285905\pi$
$457$	1.16186e12	1.24604	0.623021	+	0.782205i	$0.285905\pi$
$458$	0	0
$459$	1.26013e12	1.32513
$460$	0	0
$461$	1.63880e12	1.68994	0.844971	−	0.534812i	$-0.179617\pi$
$461$	1.63880e12	1.68994	0.844971	+	0.534812i	$0.179617\pi$
$462$	0	0
$463$	−2.10006e11	−0.212382	−0.106191	−	0.994346i	$-0.533865\pi$
$463$	−2.10006e11	−0.212382	−0.106191	+	0.994346i	$0.533865\pi$
$464$	0	0
$465$	−7.26563e10	−0.0720668
$466$	0	0
$467$	1.25155e11	0.121765	0.0608826	−	0.998145i	$-0.480608\pi$
$467$	1.25155e11	0.121765	0.0608826	+	0.998145i	$0.480608\pi$
$468$	0	0
$469$	−1.31676e12	−1.25670
$470$	0	0
$471$	−7.91867e11	−0.741409
$472$	0	0
$473$	−7.05522e11	−0.648090
$474$	0	0
$475$	1.35993e12	1.22573
$476$	0	0
$477$	−1.05094e12	−0.929489
$478$	0	0
$479$	−5.91390e11	−0.513292	−0.256646	−	0.966505i	$-0.582617\pi$
$479$	−5.91390e11	−0.513292	−0.256646	+	0.966505i	$0.582617\pi$
$480$	0	0
$481$	−9.17830e10	−0.0781825
$482$	0	0
$483$	2.24083e10	0.0187347
$484$	0	0
$485$	8.36091e10	0.0686145
$486$	0	0
$487$	−2.27814e11	−0.183527	−0.0917636	−	0.995781i	$-0.529250\pi$
$487$	−2.27814e11	−0.183527	−0.0917636	+	0.995781i	$0.529250\pi$
$488$	0	0
$489$	9.21452e11	0.728757
$490$	0	0
$491$	−2.15708e12	−1.67494	−0.837471	−	0.546482i	$-0.815967\pi$
$491$	−2.15708e12	−1.67494	−0.837471	+	0.546482i	$0.815967\pi$
$492$	0	0
$493$	1.24696e12	0.950693
$494$	0	0
$495$	9.14548e10	0.0684673
$496$	0	0
$497$	−3.78140e12	−2.78003
$498$	0	0
$499$	−9.20820e11	−0.664848	−0.332424	−	0.943130i	$-0.607867\pi$
$499$	−9.20820e11	−0.664848	−0.332424	+	0.943130i	$0.607867\pi$
$500$	0	0
$501$	−1.08446e12	−0.769029
$502$	0	0
$503$	6.63733e11	0.462314	0.231157	−	0.972916i	$-0.425749\pi$
$503$	6.63733e11	0.462314	0.231157	+	0.972916i	$0.425749\pi$
$504$	0	0
$505$	−1.99598e11	−0.136567
$506$	0	0
$507$	5.69924e10	0.0383072
$508$	0	0
$509$	−6.38733e11	−0.421783	−0.210892	−	0.977509i	$-0.567637\pi$
$509$	−6.38733e11	−0.421783	−0.210892	+	0.977509i	$0.567637\pi$
$510$	0	0
$511$	−3.60985e12	−2.34205
$512$	0	0
$513$	1.70290e12	1.08558
$514$	0	0
$515$	−2.61956e11	−0.164095
$516$	0	0
$517$	3.29641e11	0.202924
$518$	0	0
$519$	−6.00983e11	−0.363587
$520$	0	0
$521$	5.27246e10	0.0313504	0.0156752	−	0.999877i	$-0.495010\pi$
$521$	5.27246e10	0.0313504	0.0156752	+	0.999877i	$0.495010\pi$
$522$	0	0
$523$	−2.50922e11	−0.146650	−0.0733248	−	0.997308i	$-0.523361\pi$
$523$	−2.50922e11	−0.146650	−0.0733248	+	0.997308i	$0.523361\pi$
$524$	0	0
$525$	−1.61486e12	−0.927723
$526$	0	0
$527$	−3.19153e12	−1.80240
$528$	0	0
$529$	−1.80044e12	−0.999604
$530$	0	0
$531$	−5.25042e11	−0.286595
$532$	0	0
$533$	7.38169e11	0.396172
$534$	0	0
$535$	−9.11851e10	−0.0481207
$536$	0	0
$537$	−9.81900e11	−0.509545
$538$	0	0
$539$	3.76884e12	1.92335
$540$	0	0
$541$	2.07284e11	0.104035	0.0520174	−	0.998646i	$-0.483435\pi$
$541$	2.07284e11	0.104035	0.0520174	+	0.998646i	$0.483435\pi$
$542$	0	0
$543$	−3.37579e11	−0.166639
$544$	0	0
$545$	−3.57220e11	−0.173441
$546$	0	0
$547$	3.26079e12	1.55733	0.778663	−	0.627443i	$-0.215898\pi$
$547$	3.26079e12	1.55733	0.778663	+	0.627443i	$0.215898\pi$
$548$	0	0
$549$	1.32210e12	0.621137
$550$	0	0
$551$	1.68510e12	0.778832
$552$	0	0
$553$	1.44056e12	0.655039
$554$	0	0
$555$	−3.82632e10	−0.0171184
$556$	0	0
$557$	1.18725e12	0.522631	0.261316	−	0.965253i	$-0.415844\pi$
$557$	1.18725e12	0.522631	0.261316	+	0.965253i	$0.415844\pi$
$558$	0	0
$559$	5.55789e11	0.240745
$560$	0	0
$561$	−1.32484e12	−0.564715
$562$	0	0
$563$	8.57868e11	0.359859	0.179930	−	0.983679i	$-0.442413\pi$
$563$	8.57868e11	0.359859	0.179930	+	0.983679i	$0.442413\pi$
$564$	0	0
$565$	2.51707e11	0.103915
$566$	0	0
$567$	1.47768e12	0.600421
$568$	0	0
$569$	−3.71806e12	−1.48700	−0.743499	−	0.668737i	$-0.766835\pi$
$569$	−3.71806e12	−1.48700	−0.743499	+	0.668737i	$0.766835\pi$
$570$	0	0
$571$	3.06036e12	1.20479	0.602394	−	0.798199i	$-0.294214\pi$
$571$	3.06036e12	1.20479	0.602394	+	0.798199i	$0.294214\pi$
$572$	0	0
$573$	2.32259e12	0.900071
$574$	0	0
$575$	−5.13712e10	−0.0195981
$576$	0	0
$577$	−4.36544e11	−0.163959	−0.0819797	−	0.996634i	$-0.526124\pi$
$577$	−4.36544e11	−0.163959	−0.0819797	+	0.996634i	$0.526124\pi$
$578$	0	0
$579$	−3.14676e11	−0.116362
$580$	0	0
$581$	−5.14752e12	−1.87415
$582$	0	0
$583$	2.57419e12	0.922852
$584$	0	0
$585$	−7.20454e10	−0.0254334
$586$	0	0
$587$	2.99130e12	1.03989	0.519947	−	0.854199i	$-0.325952\pi$
$587$	2.99130e12	1.03989	0.519947	+	0.854199i	$0.325952\pi$
$588$	0	0
$589$	−4.31295e12	−1.47657
$590$	0	0
$591$	−1.50836e12	−0.508582
$592$	0	0
$593$	2.99710e12	0.995304	0.497652	−	0.867377i	$-0.334196\pi$
$593$	2.99710e12	0.995304	0.497652	+	0.867377i	$0.334196\pi$
$594$	0	0
$595$	1.07074e12	0.350233
$596$	0	0
$597$	1.71992e9	0.000554144 0
$598$	0	0
$599$	−2.59347e12	−0.823116	−0.411558	−	0.911384i	$-0.635015\pi$
$599$	−2.59347e12	−0.823116	−0.411558	+	0.911384i	$0.635015\pi$
$600$	0	0
$601$	3.76639e12	1.17758	0.588790	−	0.808286i	$-0.299605\pi$
$601$	3.76639e12	1.17758	0.588790	+	0.808286i	$0.299605\pi$
$602$	0	0
$603$	1.62247e12	0.499745
$604$	0	0
$605$	1.77832e11	0.0539648
$606$	0	0
$607$	−1.04286e12	−0.311802	−0.155901	−	0.987773i	$-0.549828\pi$
$607$	−1.04286e12	−0.311802	−0.155901	+	0.987773i	$0.549828\pi$
$608$	0	0
$609$	−2.00099e12	−0.589477
$610$	0	0
$611$	−2.59682e11	−0.0753799
$612$	0	0
$613$	6.63365e12	1.89749	0.948747	−	0.316035i	$-0.102352\pi$
$613$	6.63365e12	1.89749	0.948747	+	0.316035i	$0.102352\pi$
$614$	0	0
$615$	3.07734e11	0.0867436
$616$	0	0
$617$	−2.71392e12	−0.753901	−0.376951	−	0.926233i	$-0.623027\pi$
$617$	−2.71392e12	−0.753901	−0.376951	+	0.926233i	$0.623027\pi$
$618$	0	0
$619$	3.53713e12	0.968373	0.484187	−	0.874965i	$-0.339116\pi$
$619$	3.53713e12	0.968373	0.484187	+	0.874965i	$0.339116\pi$
$620$	0	0
$621$	−6.43268e10	−0.0173572
$622$	0	0
$623$	2.47784e12	0.658988
$624$	0	0
$625$	3.64537e12	0.955611
$626$	0	0
$627$	−1.79035e12	−0.462629
$628$	0	0
$629$	−1.68077e12	−0.428134
$630$	0	0
$631$	−7.28172e12	−1.82853	−0.914264	−	0.405119i	$-0.867230\pi$
$631$	−7.28172e12	−1.82853	−0.914264	+	0.405119i	$0.867230\pi$
$632$	0	0
$633$	−4.95989e11	−0.122788
$634$	0	0
$635$	2.62123e11	0.0639769
$636$	0	0
$637$	−2.96898e12	−0.714463
$638$	0	0
$639$	4.65930e12	1.10552
$640$	0	0
$641$	3.26084e12	0.762901	0.381450	−	0.924389i	$-0.375425\pi$
$641$	3.26084e12	0.762901	0.381450	+	0.924389i	$0.375425\pi$
$642$	0	0
$643$	2.90478e12	0.670137	0.335068	−	0.942194i	$-0.391241\pi$
$643$	2.90478e12	0.670137	0.335068	+	0.942194i	$0.391241\pi$
$644$	0	0
$645$	2.31702e11	0.0527121
$646$	0	0
$647$	−3.44395e12	−0.772659	−0.386330	−	0.922361i	$-0.626257\pi$
$647$	−3.44395e12	−0.772659	−0.386330	+	0.922361i	$0.626257\pi$
$648$	0	0
$649$	1.28605e12	0.284549
$650$	0	0
$651$	5.12145e12	1.11758
$652$	0	0
$653$	−1.74151e12	−0.374815	−0.187407	−	0.982282i	$-0.560008\pi$
$653$	−1.74151e12	−0.374815	−0.187407	+	0.982282i	$0.560008\pi$
$654$	0	0
$655$	−2.72210e11	−0.0577854
$656$	0	0
$657$	4.44793e12	0.931351
$658$	0	0
$659$	−4.57973e12	−0.945922	−0.472961	−	0.881083i	$-0.656815\pi$
$659$	−4.57973e12	−0.945922	−0.472961	+	0.881083i	$0.656815\pi$
$660$	0	0
$661$	−6.64788e12	−1.35449	−0.677246	−	0.735756i	$-0.736827\pi$
$661$	−6.64788e12	−1.35449	−0.677246	+	0.735756i	$0.736827\pi$
$662$	0	0
$663$	1.04367e12	0.209774
$664$	0	0
$665$	1.44696e12	0.286920
$666$	0	0
$667$	−6.36545e10	−0.0124527
$668$	0	0
$669$	−3.20777e12	−0.619136
$670$	0	0
$671$	−3.23837e12	−0.616702
$672$	0	0
$673$	5.04803e12	0.948536	0.474268	−	0.880381i	$-0.342713\pi$
$673$	5.04803e12	0.948536	0.474268	+	0.880381i	$0.342713\pi$
$674$	0	0
$675$	4.63574e12	0.859513
$676$	0	0
$677$	1.60646e12	0.293914	0.146957	−	0.989143i	$-0.453052\pi$
$677$	1.60646e12	0.293914	0.146957	+	0.989143i	$0.453052\pi$
$678$	0	0
$679$	−5.89349e12	−1.06404
$680$	0	0
$681$	6.76010e9	0.00120446
$682$	0	0
$683$	−7.46072e12	−1.31186	−0.655930	−	0.754821i	$-0.727724\pi$
$683$	−7.46072e12	−1.31186	−0.655930	+	0.754821i	$0.727724\pi$
$684$	0	0
$685$	1.21494e12	0.210838
$686$	0	0
$687$	3.17090e12	0.543096
$688$	0	0
$689$	−2.02787e12	−0.342810
$690$	0	0
$691$	−5.85828e12	−0.977506	−0.488753	−	0.872422i	$-0.662548\pi$
$691$	−5.85828e12	−0.977506	−0.488753	+	0.872422i	$0.662548\pi$
$692$	0	0
$693$	−6.44653e12	−1.06176
$694$	0	0
$695$	1.14732e12	0.186532
$696$	0	0
$697$	1.35177e13	2.16947
$698$	0	0
$699$	−9.10470e11	−0.144251
$700$	0	0
$701$	6.30209e12	0.985720	0.492860	−	0.870109i	$-0.335951\pi$
$701$	6.30209e12	0.985720	0.492860	+	0.870109i	$0.335951\pi$
$702$	0	0
$703$	−2.27134e12	−0.350738
$704$	0	0
$705$	−1.08258e11	−0.0165048
$706$	0	0
$707$	1.40694e13	2.11781
$708$	0	0
$709$	9.81972e12	1.45946	0.729728	−	0.683737i	$-0.239647\pi$
$709$	9.81972e12	1.45946	0.729728	+	0.683737i	$0.239647\pi$
$710$	0	0
$711$	−1.77500e12	−0.260486
$712$	0	0
$713$	1.62921e11	0.0236088
$714$	0	0
$715$	1.76470e11	0.0252518
$716$	0	0
$717$	−6.28228e11	−0.0887731
$718$	0	0
$719$	6.50102e12	0.907197	0.453599	−	0.891206i	$-0.350140\pi$
$719$	6.50102e12	0.907197	0.453599	+	0.891206i	$0.350140\pi$
$720$	0	0
$721$	1.84649e13	2.54471
$722$	0	0
$723$	5.57289e12	0.758504
$724$	0	0
$725$	4.58729e12	0.616645
$726$	0	0
$727$	−1.35766e13	−1.80255	−0.901273	−	0.433252i	$-0.857366\pi$
$727$	−1.35766e13	−1.80255	−0.901273	+	0.433252i	$0.857366\pi$
$728$	0	0
$729$	1.49253e12	0.195727
$730$	0	0
$731$	1.01778e13	1.31834
$732$	0	0
$733$	7.63316e12	0.976644	0.488322	−	0.872664i	$-0.337609\pi$
$733$	7.63316e12	0.976644	0.488322	+	0.872664i	$0.337609\pi$
$734$	0	0
$735$	−1.23773e12	−0.156435
$736$	0	0
$737$	−3.97411e12	−0.496176
$738$	0	0
$739$	−8.49835e12	−1.04818	−0.524089	−	0.851664i	$-0.675594\pi$
$739$	−8.49835e12	−1.04818	−0.524089	+	0.851664i	$0.675594\pi$
$740$	0	0
$741$	1.41038e12	0.171852
$742$	0	0
$743$	−2.39332e12	−0.288105	−0.144053	−	0.989570i	$-0.546013\pi$
$743$	−2.39332e12	−0.288105	−0.144053	+	0.989570i	$0.546013\pi$
$744$	0	0
$745$	1.95772e12	0.232835
$746$	0	0
$747$	6.34258e12	0.745287
$748$	0	0
$749$	6.42752e12	0.746234
$750$	0	0
$751$	−5.80031e12	−0.665383	−0.332691	−	0.943036i	$-0.607957\pi$
$751$	−5.80031e12	−0.665383	−0.332691	+	0.943036i	$0.607957\pi$
$752$	0	0
$753$	−4.92748e12	−0.558532
$754$	0	0
$755$	−1.34831e12	−0.151018
$756$	0	0
$757$	4.07992e12	0.451565	0.225783	−	0.974178i	$-0.427506\pi$
$757$	4.07992e12	0.451565	0.225783	+	0.974178i	$0.427506\pi$
$758$	0	0
$759$	6.76301e10	0.00739694
$760$	0	0
$761$	−1.34251e13	−1.45107	−0.725534	−	0.688186i	$-0.758407\pi$
$761$	−1.34251e13	−1.45107	−0.725534	+	0.688186i	$0.758407\pi$
$762$	0	0
$763$	2.51800e13	2.68964
$764$	0	0
$765$	−1.31932e12	−0.139276
$766$	0	0
$767$	−1.01311e12	−0.105701
$768$	0	0
$769$	−1.27465e13	−1.31438	−0.657191	−	0.753724i	$-0.728255\pi$
$769$	−1.27465e13	−1.31438	−0.657191	+	0.753724i	$0.728255\pi$
$770$	0	0
$771$	−2.52977e12	−0.257832
$772$	0	0
$773$	−1.14502e13	−1.15347	−0.576734	−	0.816932i	$-0.695673\pi$
$773$	−1.14502e13	−1.15347	−0.576734	+	0.816932i	$0.695673\pi$
$774$	0	0
$775$	−1.17410e13	−1.16909
$776$	0	0
$777$	2.69712e12	0.265464
$778$	0	0
$779$	1.82674e13	1.77729
$780$	0	0
$781$	−1.14126e13	−1.09763
$782$	0	0
$783$	5.74419e12	0.546136
$784$	0	0
$785$	1.93155e12	0.181548
$786$	0	0
$787$	−3.80733e12	−0.353781	−0.176890	−	0.984231i	$-0.556604\pi$
$787$	−3.80733e12	−0.353781	−0.176890	+	0.984231i	$0.556604\pi$
$788$	0	0
$789$	−1.13831e12	−0.104572
$790$	0	0
$791$	−1.77425e13	−1.61146
$792$	0	0
$793$	2.55109e12	0.229085
$794$	0	0
$795$	−8.45395e11	−0.0750598
$796$	0	0
$797$	−3.60696e12	−0.316650	−0.158325	−	0.987387i	$-0.550609\pi$
$797$	−3.60696e12	−0.316650	−0.158325	+	0.987387i	$0.550609\pi$
$798$	0	0
$799$	−4.75539e12	−0.412787
$800$	0	0
$801$	−3.05310e12	−0.262057
$802$	0	0
$803$	−1.08948e13	−0.924701
$804$	0	0
$805$	−5.46589e10	−0.00458754
$806$	0	0
$807$	6.19210e12	0.513933
$808$	0	0
$809$	2.73587e12	0.224557	0.112279	−	0.993677i	$-0.464185\pi$
$809$	2.73587e12	0.224557	0.112279	+	0.993677i	$0.464185\pi$
$810$	0	0
$811$	3.38202e12	0.274526	0.137263	−	0.990535i	$-0.456170\pi$
$811$	3.38202e12	0.274526	0.137263	+	0.990535i	$0.456170\pi$
$812$	0	0
$813$	4.44886e12	0.357142
$814$	0	0
$815$	−2.24763e12	−0.178450
$816$	0	0
$817$	1.37540e13	1.08002
$818$	0	0
$819$	5.07838e12	0.394410
$820$	0	0
$821$	2.30840e13	1.77324	0.886620	−	0.462499i	$-0.153047\pi$
$821$	2.30840e13	1.77324	0.886620	+	0.462499i	$0.153047\pi$
$822$	0	0
$823$	−2.30292e12	−0.174977	−0.0874883	−	0.996166i	$-0.527884\pi$
$823$	−2.30292e12	−0.174977	−0.0874883	+	0.996166i	$0.527884\pi$
$824$	0	0
$825$	−4.87379e12	−0.366289
$826$	0	0
$827$	−1.25394e13	−0.932188	−0.466094	−	0.884735i	$-0.654339\pi$
$827$	−1.25394e13	−0.932188	−0.466094	+	0.884735i	$0.654339\pi$
$828$	0	0
$829$	2.52253e13	1.85498	0.927492	−	0.373843i	$-0.121960\pi$
$829$	2.52253e13	1.85498	0.927492	+	0.373843i	$0.121960\pi$
$830$	0	0
$831$	8.33901e12	0.606610
$832$	0	0
$833$	−5.43691e13	−3.91246
$834$	0	0
$835$	2.64524e12	0.188311
$836$	0	0
$837$	−1.47020e13	−1.03541
$838$	0	0
$839$	1.03503e13	0.721146	0.360573	−	0.932731i	$-0.382581\pi$
$839$	1.03503e13	0.721146	0.360573	+	0.932731i	$0.382581\pi$
$840$	0	0
$841$	−8.82300e12	−0.608183
$842$	0	0
$843$	−3.95962e12	−0.270041
$844$	0	0
$845$	−1.39018e11	−0.00938025
$846$	0	0
$847$	−1.25351e13	−0.836862
$848$	0	0
$849$	1.03759e13	0.685394
$850$	0	0
$851$	8.57996e10	0.00560793
$852$	0	0
$853$	5.58791e12	0.361392	0.180696	−	0.983539i	$-0.442165\pi$
$853$	5.58791e12	0.361392	0.180696	+	0.983539i	$0.442165\pi$
$854$	0	0
$855$	−1.78290e12	−0.114098
$856$	0	0
$857$	−2.86481e13	−1.81418	−0.907092	−	0.420932i	$-0.861703\pi$
$857$	−2.86481e13	−1.81418	−0.907092	+	0.420932i	$0.861703\pi$
$858$	0	0
$859$	−1.08878e13	−0.682290	−0.341145	−	0.940011i	$-0.610815\pi$
$859$	−1.08878e13	−0.682290	−0.341145	+	0.940011i	$0.610815\pi$
$860$	0	0
$861$	−2.16917e13	−1.34518
$862$	0	0
$863$	1.56005e11	0.00957393	0.00478697	−	0.999989i	$-0.498476\pi$
$863$	1.56005e11	0.00957393	0.00478697	+	0.999989i	$0.498476\pi$
$864$	0	0
$865$	1.46594e12	0.0890312
$866$	0	0
$867$	1.08267e13	0.650744
$868$	0	0
$869$	4.34772e12	0.258626
$870$	0	0
$871$	3.13069e12	0.184314
$872$	0	0
$873$	7.26174e12	0.423133
$874$	0	0
$875$	7.93750e12	0.457770
$876$	0	0
$877$	4.54034e12	0.259173	0.129587	−	0.991568i	$-0.458635\pi$
$877$	4.54034e12	0.259173	0.129587	+	0.991568i	$0.458635\pi$
$878$	0	0
$879$	1.38543e13	0.782771
$880$	0	0
$881$	1.22592e13	0.685598	0.342799	−	0.939409i	$-0.388625\pi$
$881$	1.22592e13	0.685598	0.342799	+	0.939409i	$0.388625\pi$
$882$	0	0
$883$	1.15154e13	0.637462	0.318731	−	0.947845i	$-0.396743\pi$
$883$	1.15154e13	0.637462	0.318731	+	0.947845i	$0.396743\pi$
$884$	0	0
$885$	−4.22354e11	−0.0231437
$886$	0	0
$887$	3.41697e13	1.85347	0.926735	−	0.375716i	$-0.122603\pi$
$887$	3.41697e13	1.85347	0.926735	+	0.375716i	$0.122603\pi$
$888$	0	0
$889$	−1.84767e13	−0.992125
$890$	0	0
$891$	4.45976e12	0.237062
$892$	0	0
$893$	−6.42630e12	−0.338165
$894$	0	0
$895$	2.39508e12	0.124772
$896$	0	0
$897$	−5.32770e10	−0.00274773
$898$	0	0
$899$	−1.45483e13	−0.742839
$900$	0	0
$901$	−3.71352e13	−1.87726
$902$	0	0
$903$	−1.63323e13	−0.817436
$904$	0	0
$905$	8.23432e11	0.0408046
$906$	0	0
$907$	1.30313e13	0.639373	0.319687	−	0.947523i	$-0.396422\pi$
$907$	1.30313e13	0.639373	0.319687	+	0.947523i	$0.396422\pi$
$908$	0	0
$909$	−1.73358e13	−0.842181
$910$	0	0
$911$	−5.90209e12	−0.283905	−0.141953	−	0.989873i	$-0.545338\pi$
$911$	−5.90209e12	−0.283905	−0.141953	+	0.989873i	$0.545338\pi$
$912$	0	0
$913$	−1.55357e13	−0.739965
$914$	0	0
$915$	1.06352e12	0.0501592
$916$	0	0
$917$	1.91877e13	0.896110
$918$	0	0
$919$	−8.87655e11	−0.0410511	−0.0205255	−	0.999789i	$-0.506534\pi$
$919$	−8.87655e11	−0.0410511	−0.0205255	+	0.999789i	$0.506534\pi$
$920$	0	0
$921$	1.51430e13	0.693496
$922$	0	0
$923$	8.99051e12	0.407734
$924$	0	0
$925$	−6.18319e12	−0.277699
$926$	0	0
$927$	−2.27518e13	−1.01195
$928$	0	0
$929$	−5.69435e12	−0.250826	−0.125413	−	0.992105i	$-0.540026\pi$
$929$	−5.69435e12	−0.250826	−0.125413	+	0.992105i	$0.540026\pi$
$930$	0	0
$931$	−7.34729e13	−3.20519
$932$	0	0
$933$	1.75018e13	0.756164
$934$	0	0
$935$	3.23158e12	0.138281
$936$	0	0
$937$	−3.18002e13	−1.34773	−0.673864	−	0.738856i	$-0.735366\pi$
$937$	−3.18002e13	−1.34773	−0.673864	+	0.738856i	$0.735366\pi$
$938$	0	0
$939$	−1.91025e13	−0.801853
$940$	0	0
$941$	1.56601e12	0.0651090	0.0325545	−	0.999470i	$-0.489636\pi$
$941$	1.56601e12	0.0651090	0.0325545	+	0.999470i	$0.489636\pi$
$942$	0	0
$943$	−6.90048e11	−0.0284169
$944$	0	0
$945$	4.93242e12	0.201195
$946$	0	0
$947$	3.79230e13	1.53224	0.766122	−	0.642695i	$-0.222184\pi$
$947$	3.79230e13	1.53224	0.766122	+	0.642695i	$0.222184\pi$
$948$	0	0
$949$	8.58264e12	0.343497
$950$	0	0
$951$	−9.77598e12	−0.387568
$952$	0	0
$953$	1.56014e13	0.612698	0.306349	−	0.951919i	$-0.400893\pi$
$953$	1.56014e13	0.612698	0.306349	+	0.951919i	$0.400893\pi$
$954$	0	0
$955$	−5.66533e12	−0.220399
$956$	0	0
$957$	−6.03916e12	−0.232741
$958$	0	0
$959$	−8.56397e13	−3.26958
$960$	0	0
$961$	1.07963e13	0.408336
$962$	0	0
$963$	−7.91975e12	−0.296752
$964$	0	0
$965$	7.67566e11	0.0284933
$966$	0	0
$967$	1.70795e13	0.628138	0.314069	−	0.949400i	$-0.398308\pi$
$967$	1.70795e13	0.628138	0.314069	+	0.949400i	$0.398308\pi$
$968$	0	0
$969$	2.58275e13	0.941076
$970$	0	0
$971$	−2.52915e12	−0.0913035	−0.0456517	−	0.998957i	$-0.514536\pi$
$971$	−2.52915e12	−0.0913035	−0.0456517	+	0.998957i	$0.514536\pi$
$972$	0	0
$973$	−8.08733e13	−2.89266
$974$	0	0
$975$	3.83943e12	0.136065
$976$	0	0
$977$	−4.20451e13	−1.47635	−0.738176	−	0.674609i	$-0.764312\pi$
$977$	−4.20451e13	−1.47635	−0.738176	+	0.674609i	$0.764312\pi$
$978$	0	0
$979$	7.47834e12	0.260186
$980$	0	0
$981$	−3.10258e13	−1.06958
$982$	0	0
$983$	8.07032e12	0.275677	0.137838	−	0.990455i	$-0.455985\pi$
$983$	8.07032e12	0.275677	0.137838	+	0.990455i	$0.455985\pi$
$984$	0	0
$985$	3.67923e12	0.124536
$986$	0	0
$987$	7.63097e12	0.255948
$988$	0	0
$989$	−5.19557e11	−0.0172683
$990$	0	0
$991$	−1.93018e13	−0.635721	−0.317861	−	0.948137i	$-0.602964\pi$
$991$	−1.93018e13	−0.635721	−0.317861	+	0.948137i	$0.602964\pi$
$992$	0	0
$993$	−2.17750e13	−0.710700
$994$	0	0
$995$	−4.19527e9	−0.000135693 0
$996$	0	0
$997$	1.74755e13	0.560145	0.280073	−	0.959979i	$-0.409641\pi$
$997$	1.74755e13	0.560145	0.280073	+	0.959979i	$0.409641\pi$
$998$	0	0
$999$	−7.74256e12	−0.245946

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	208.10.a.l.1.5		7
4.3	odd	2		104.10.a.c.1.3	✓	7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
104.10.a.c.1.3	✓	7	4.3	odd	2
208.10.a.l.1.5		7	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 208 = 2^{4} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	208.a (trivial)

\(f(q)\)	\(=\)	\(q+69.8667 q^{3} -170.421 q^{5} +12012.7 q^{7} -14801.7 q^{9} +36255.5 q^{11} -28561.0 q^{13} -11906.7 q^{15} -523020. q^{17} -706794. q^{19} +839290. q^{21} +26699.1 q^{23} -1.92408e6 q^{25} -2.40933e6 q^{27} -2.38415e6 q^{29} +6.10212e6 q^{31} +2.53305e6 q^{33} -2.04722e6 q^{35} +3.21358e6 q^{37} -1.99546e6 q^{39} -2.58454e7 q^{41} -1.94597e7 q^{43} +2.52251e6 q^{45} +9.09217e6 q^{47} +1.03952e8 q^{49} -3.65417e7 q^{51} +7.10014e7 q^{53} -6.17869e6 q^{55} -4.93814e7 q^{57} +3.54719e7 q^{59} -8.93209e7 q^{61} -1.77808e8 q^{63} +4.86739e6 q^{65} -1.09614e8 q^{67} +1.86538e6 q^{69} -3.14783e8 q^{71} -3.00502e8 q^{73} -1.34429e8 q^{75} +4.35528e8 q^{77} +1.19919e8 q^{79} +1.23009e8 q^{81} -4.28505e8 q^{83} +8.91336e7 q^{85} -1.66572e8 q^{87} +2.06268e8 q^{89} -3.43096e8 q^{91} +4.26335e8 q^{93} +1.20453e8 q^{95} -4.90604e8 q^{97} -5.36641e8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q - 13 q^{3} - 801 q^{5} + 1091 q^{7} + 87864 q^{9} - 32218 q^{11} - 199927 q^{13} - 58323 q^{15} - 870531 q^{17} + 128950 q^{19} - 719663 q^{21} - 2198844 q^{23} + 992010 q^{25} - 5441383 q^{27} + 6327710 q^{29}+ \cdots - 5204667600 q^{99}+O(q^{100}) \) 7 * q - 13 * q^3 - 801 * q^5 + 1091 * q^7 + 87864 * q^9 - 32218 * q^11 - 199927 * q^13 - 58323 * q^15 - 870531 * q^17 + 128950 * q^19 - 719663 * q^21 - 2198844 * q^23 + 992010 * q^25 - 5441383 * q^27 + 6327710 * q^29 + 1660376 * q^31 + 14199598 * q^33 + 19231769 * q^35 + 5058787 * q^37 + 371293 * q^39 + 14779748 * q^41 - 9405075 * q^43 + 56644022 * q^45 - 133415721 * q^47 + 44544988 * q^49 - 132066169 * q^51 + 43962424 * q^53 - 191344746 * q^55 + 138963078 * q^57 - 22370274 * q^59 + 23851748 * q^61 - 780720706 * q^63 + 22877361 * q^65 - 12058786 * q^67 + 1036453420 * q^69 - 864303791 * q^71 + 670491202 * q^73 - 693324044 * q^75 + 841052238 * q^77 - 1019547572 * q^79 + 900194079 * q^81 - 1345041700 * q^83 + 824066375 * q^85 - 2589172074 * q^87 + 1409158546 * q^89 - 31160051 * q^91 - 445200892 * q^93 - 3285213058 * q^95 + 320206906 * q^97 - 5204667600 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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