LMFDB - Embedded newform 208.10.a.j.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [208,10,Mod(1,208)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

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")

//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: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$107.127453922$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 69754x^{4} - 2752492x^{3} + 1089377733x^{2} + 50183965132x - 2195812679340 $ x^6 - 69754x^4 - 2752492x^3 + 1089377733x^2 + 50183965132x - 2195812679340
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{12}\cdot 3 $
Twist minimal:	no (minimal twist has level 104)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-162.332$ 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)

$f(q)$	$=$	$q-172.332 q^{3} +107.467 q^{5} +615.682 q^{7} +10015.3 q^{9} +40196.2 q^{11} -28561.0 q^{13} -18520.0 q^{15} -489315. q^{17} -730371. q^{19} -106102. q^{21} +2.11920e6 q^{23} -1.94158e6 q^{25} +1.66605e6 q^{27} +5.06021e6 q^{29} +6.84851e6 q^{31} -6.92710e6 q^{33} +66165.5 q^{35} -8.66721e6 q^{37} +4.92197e6 q^{39} +3.24853e7 q^{41} -654425. q^{43} +1.07632e6 q^{45} -1.57092e7 q^{47} -3.99745e7 q^{49} +8.43247e7 q^{51} +2.50249e7 q^{53} +4.31977e6 q^{55} +1.25866e8 q^{57} +3.01805e7 q^{59} -1.22605e7 q^{61} +6.16626e6 q^{63} -3.06937e6 q^{65} -8.97056e7 q^{67} -3.65207e8 q^{69} +3.12230e8 q^{71} +6.69648e7 q^{73} +3.34596e8 q^{75} +2.47481e7 q^{77} +2.97738e8 q^{79} -4.84245e8 q^{81} +6.01224e8 q^{83} -5.25853e7 q^{85} -8.72037e8 q^{87} -9.77021e8 q^{89} -1.75845e7 q^{91} -1.18022e9 q^{93} -7.84908e7 q^{95} -1.04646e9 q^{97} +4.02578e8 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 60 q^{3} + 176 q^{5} - 3416 q^{7} + 22010 q^{9} + 101316 q^{11} - 171366 q^{13} - 257460 q^{15} - 170304 q^{17} + 92084 q^{19} - 424004 q^{21} - 2369944 q^{23} + 2043598 q^{25} + 6428196 q^{27} - 9021404 q^{29}+ \cdots - 282095980 q^{99}+O(q^{100}) $ 6 * q - 60 * q^3 + 176 * q^5 - 3416 * q^7 + 22010 * q^9 + 101316 * q^11 - 171366 * q^13 - 257460 * q^15 - 170304 * q^17 + 92084 * q^19 - 424004 * q^21 - 2369944 * q^23 + 2043598 * q^25 + 6428196 * q^27 - 9021404 * q^29 + 9894388 * q^31 + 6717200 * q^33 + 1910916 * q^35 + 2974376 * q^37 + 1713660 * q^39 - 2977668 * q^41 + 31585340 * q^43 - 99427116 * q^45 + 1766688 * q^47 - 25397750 * q^49 + 68102932 * q^51 - 137836444 * q^53 + 64702520 * q^55 - 253569848 * q^57 + 128523972 * q^59 + 33822404 * q^61 + 359085076 * q^63 - 5026736 * q^65 + 618333476 * q^67 - 837582264 * q^69 + 52534456 * q^71 - 667586460 * q^73 + 940557360 * q^75 - 944797672 * q^77 + 399417496 * q^79 - 821518498 * q^81 + 467205356 * q^83 - 2583200620 * q^85 + 219585328 * q^87 - 1630231268 * q^89 + 97564376 * q^91 - 585888632 * q^93 + 3079232768 * q^95 - 3266492188 * q^97 - 282095980 * 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
$3$	−172.332	−1.22834	−0.614172	−	0.789172i	$-0.710510\pi$
$3$	−172.332	−1.22834	−0.614172	+	0.789172i	$0.710510\pi$
$4$	0	0
$5$	107.467	0.0768971	0.0384486	−	0.999261i	$-0.487758\pi$
$5$	107.467	0.0768971	0.0384486	+	0.999261i	$0.487758\pi$
$6$	0	0
$7$	615.682	0.0969204	0.0484602	−	0.998825i	$-0.484569\pi$
$7$	615.682	0.0969204	0.0484602	+	0.998825i	$0.484569\pi$
$8$	0	0
$9$	10015.3	0.508831
$10$	0	0
$11$	40196.2	0.827786	0.413893	−	0.910326i	$-0.364169\pi$
$11$	40196.2	0.827786	0.413893	+	0.910326i	$0.364169\pi$
$12$	0	0
$13$	−28561.0	−0.277350
$13$	−28561.0	−0.277350
$14$	0	0
$15$	−18520.0	−0.0944562
$16$	0	0
$17$	−489315.	−1.42092	−0.710458	−	0.703739i	$-0.751512\pi$
$17$	−489315.	−1.42092	−0.710458	+	0.703739i	$0.751512\pi$
$18$	0	0
$19$	−730371.	−1.28574	−0.642869	−	0.765976i	$-0.722256\pi$
$19$	−730371.	−1.28574	−0.642869	+	0.765976i	$0.722256\pi$
$20$	0	0
$21$	−106102.	−0.119052
$22$	0	0
$23$	2.11920e6	1.57906	0.789528	−	0.613715i	$-0.210326\pi$
$23$	2.11920e6	1.57906	0.789528	+	0.613715i	$0.210326\pi$
$24$	0	0
$25$	−1.94158e6	−0.994087
$26$	0	0
$27$	1.66605e6	0.603325
$28$	0	0
$29$	5.06021e6	1.32855	0.664275	−	0.747488i	$-0.268740\pi$
$29$	5.06021e6	1.32855	0.664275	+	0.747488i	$0.268740\pi$
$30$	0	0
$31$	6.84851e6	1.33189	0.665945	−	0.746001i	$-0.268029\pi$
$31$	6.84851e6	1.33189	0.665945	+	0.746001i	$0.268029\pi$
$32$	0	0
$33$	−6.92710e6	−1.01681
$34$	0	0
$35$	66165.5	0.00745290
$36$	0	0
$37$	−8.66721e6	−0.760276	−0.380138	−	0.924930i	$-0.624124\pi$
$37$	−8.66721e6	−0.760276	−0.380138	+	0.924930i	$0.624124\pi$
$38$	0	0
$39$	4.92197e6	0.340682
$40$	0	0
$41$	3.24853e7	1.79539	0.897697	−	0.440614i	$-0.145239\pi$
$41$	3.24853e7	1.79539	0.897697	+	0.440614i	$0.145239\pi$
$42$	0	0
$43$	−654425.	−0.0291912	−0.0145956	−	0.999893i	$-0.504646\pi$
$43$	−654425.	−0.0291912	−0.0145956	+	0.999893i	$0.504646\pi$
$44$	0	0
$45$	1.07632e6	0.0391277
$46$	0	0
$47$	−1.57092e7	−0.469586	−0.234793	−	0.972045i	$-0.575441\pi$
$47$	−1.57092e7	−0.469586	−0.234793	+	0.972045i	$0.575441\pi$
$48$	0	0
$49$	−3.99745e7	−0.990606
$50$	0	0
$51$	8.43247e7	1.74538
$52$	0	0
$53$	2.50249e7	0.435643	0.217822	−	0.975989i	$-0.430105\pi$
$53$	2.50249e7	0.435643	0.217822	+	0.975989i	$0.430105\pi$
$54$	0	0
$55$	4.31977e6	0.0636544
$56$	0	0
$57$	1.25866e8	1.57933
$58$	0	0
$59$	3.01805e7	0.324260	0.162130	−	0.986769i	$-0.448164\pi$
$59$	3.01805e7	0.324260	0.162130	+	0.986769i	$0.448164\pi$
$60$	0	0
$61$	−1.22605e7	−0.113377	−0.0566885	−	0.998392i	$-0.518054\pi$
$61$	−1.22605e7	−0.113377	−0.0566885	+	0.998392i	$0.518054\pi$
$62$	0	0
$63$	6.16626e6	0.0493161
$64$	0	0
$65$	−3.06937e6	−0.0213274
$66$	0	0
$67$	−8.97056e7	−0.543855	−0.271927	−	0.962318i	$-0.587661\pi$
$67$	−8.97056e7	−0.543855	−0.271927	+	0.962318i	$0.587661\pi$
$68$	0	0
$69$	−3.65207e8	−1.93962
$70$	0	0
$71$	3.12230e8	1.45818	0.729092	−	0.684415i	$-0.239942\pi$
$71$	3.12230e8	1.45818	0.729092	+	0.684415i	$0.239942\pi$
$72$	0	0
$73$	6.69648e7	0.275990	0.137995	−	0.990433i	$-0.455934\pi$
$73$	6.69648e7	0.275990	0.137995	+	0.990433i	$0.455934\pi$
$74$	0	0
$75$	3.34596e8	1.22108
$76$	0	0
$77$	2.47481e7	0.0802294
$78$	0	0
$79$	2.97738e8	0.860028	0.430014	−	0.902822i	$-0.358509\pi$
$79$	2.97738e8	0.860028	0.430014	+	0.902822i	$0.358509\pi$
$80$	0	0
$81$	−4.84245e8	−1.24992
$82$	0	0
$83$	6.01224e8	1.39054	0.695272	−	0.718747i	$-0.255284\pi$
$83$	6.01224e8	1.39054	0.695272	+	0.718747i	$0.255284\pi$
$84$	0	0
$85$	−5.25853e7	−0.109264
$86$	0	0
$87$	−8.72037e8	−1.63192
$88$	0	0
$89$	−9.77021e8	−1.65063	−0.825313	−	0.564675i	$-0.809002\pi$
$89$	−9.77021e8	−1.65063	−0.825313	+	0.564675i	$0.809002\pi$
$90$	0	0
$91$	−1.75845e7	−0.0268809
$92$	0	0
$93$	−1.18022e9	−1.63602
$94$	0	0
$95$	−7.84908e7	−0.0988696
$96$	0	0
$97$	−1.04646e9	−1.20019	−0.600093	−	0.799930i	$-0.704870\pi$
$97$	−1.04646e9	−1.20019	−0.600093	+	0.799930i	$0.704870\pi$
$98$	0	0
$99$	4.02578e8	0.421204
$100$	0	0
$101$	1.35004e9	1.29092	0.645460	−	0.763794i	$-0.276666\pi$
$101$	1.35004e9	1.29092	0.645460	+	0.763794i	$0.276666\pi$
$102$	0	0
$103$	−4.91015e8	−0.429860	−0.214930	−	0.976629i	$-0.568952\pi$
$103$	−4.91015e8	−0.429860	−0.214930	+	0.976629i	$0.568952\pi$
$104$	0	0
$105$	−1.14024e7	−0.00915474
$106$	0	0
$107$	−2.50993e9	−1.85112	−0.925560	−	0.378600i	$-0.876406\pi$
$107$	−2.50993e9	−1.85112	−0.925560	+	0.378600i	$0.876406\pi$
$108$	0	0
$109$	−2.02473e8	−0.137388	−0.0686939	−	0.997638i	$-0.521883\pi$
$109$	−2.02473e8	−0.137388	−0.0686939	+	0.997638i	$0.521883\pi$
$110$	0	0
$111$	1.49364e9	0.933881
$112$	0	0
$113$	−1.76686e9	−1.01941	−0.509705	−	0.860349i	$-0.670246\pi$
$113$	−1.76686e9	−1.01941	−0.509705	+	0.860349i	$0.670246\pi$
$114$	0	0
$115$	2.27744e8	0.121425
$116$	0	0
$117$	−2.86048e8	−0.141124
$118$	0	0
$119$	−3.01263e8	−0.137716
$120$	0	0
$121$	−7.42211e8	−0.314770
$122$	0	0
$123$	−5.59826e9	−2.20536
$124$	0	0
$125$	−4.18552e8	−0.153340
$126$	0	0
$127$	−2.47555e9	−0.844413	−0.422207	−	0.906500i	$-0.638744\pi$
$127$	−2.47555e9	−0.844413	−0.422207	+	0.906500i	$0.638744\pi$
$128$	0	0
$129$	1.12778e8	0.0358569
$130$	0	0
$131$	−1.70806e9	−0.506736	−0.253368	−	0.967370i	$-0.581538\pi$
$131$	−1.70806e9	−0.506736	−0.253368	+	0.967370i	$0.581538\pi$
$132$	0	0
$133$	−4.49677e8	−0.124614
$134$	0	0
$135$	1.79045e8	0.0463939
$136$	0	0
$137$	−1.05278e9	−0.255327	−0.127664	−	0.991818i	$-0.540748\pi$
$137$	−1.05278e9	−0.255327	−0.127664	+	0.991818i	$0.540748\pi$
$138$	0	0
$139$	−3.23568e8	−0.0735189	−0.0367595	−	0.999324i	$-0.511704\pi$
$139$	−3.23568e8	−0.0735189	−0.0367595	+	0.999324i	$0.511704\pi$
$140$	0	0
$141$	2.70721e9	0.576813
$142$	0	0
$143$	−1.14804e9	−0.229587
$144$	0	0
$145$	5.43806e8	0.102162
$146$	0	0
$147$	6.88889e9	1.21681
$148$	0	0
$149$	−6.20510e9	−1.03136	−0.515680	−	0.856781i	$-0.672461\pi$
$149$	−6.20510e9	−1.03136	−0.515680	+	0.856781i	$0.672461\pi$
$150$	0	0
$151$	−4.68351e9	−0.733120	−0.366560	−	0.930394i	$-0.619465\pi$
$151$	−4.68351e9	−0.733120	−0.366560	+	0.930394i	$0.619465\pi$
$152$	0	0
$153$	−4.90065e9	−0.723007
$154$	0	0
$155$	7.35988e8	0.102418
$156$	0	0
$157$	7.87350e9	1.03423	0.517117	−	0.855914i	$-0.327005\pi$
$157$	7.87350e9	1.03423	0.517117	+	0.855914i	$0.327005\pi$
$158$	0	0
$159$	−4.31259e9	−0.535120
$160$	0	0
$161$	1.30476e9	0.153043
$162$	0	0
$163$	−8.45414e9	−0.938048	−0.469024	−	0.883185i	$-0.655394\pi$
$163$	−8.45414e9	−0.938048	−0.469024	+	0.883185i	$0.655394\pi$
$164$	0	0
$165$	−7.44434e8	−0.0781895
$166$	0	0
$167$	−1.56518e9	−0.155719	−0.0778595	−	0.996964i	$-0.524809\pi$
$167$	−1.56518e9	−0.155719	−0.0778595	+	0.996964i	$0.524809\pi$
$168$	0	0
$169$	8.15731e8	0.0769231
$170$	0	0
$171$	−7.31491e9	−0.654224
$172$	0	0
$173$	2.13868e10	1.81525	0.907627	−	0.419777i	$-0.137892\pi$
$173$	2.13868e10	1.81525	0.907627	+	0.419777i	$0.137892\pi$
$174$	0	0
$175$	−1.19539e9	−0.0963473
$176$	0	0
$177$	−5.20107e9	−0.398303
$178$	0	0
$179$	1.61808e9	0.117804	0.0589022	−	0.998264i	$-0.481240\pi$
$179$	1.61808e9	0.117804	0.0589022	+	0.998264i	$0.481240\pi$
$180$	0	0
$181$	−6.97443e9	−0.483009	−0.241505	−	0.970400i	$-0.577641\pi$
$181$	−6.97443e9	−0.483009	−0.241505	+	0.970400i	$0.577641\pi$
$182$	0	0
$183$	2.11288e9	0.139266
$184$	0	0
$185$	−9.31439e8	−0.0584631
$186$	0	0
$187$	−1.96686e10	−1.17622
$188$	0	0
$189$	1.02576e9	0.0584745
$190$	0	0
$191$	−2.28374e10	−1.24164	−0.620821	−	0.783953i	$-0.713200\pi$
$191$	−2.28374e10	−1.24164	−0.620821	+	0.783953i	$0.713200\pi$
$192$	0	0
$193$	−2.79376e10	−1.44938	−0.724689	−	0.689077i	$-0.758016\pi$
$193$	−2.79376e10	−1.44938	−0.724689	+	0.689077i	$0.758016\pi$
$194$	0	0
$195$	5.28950e8	0.0261974
$196$	0	0
$197$	2.59082e10	1.22557	0.612786	−	0.790249i	$-0.290049\pi$
$197$	2.59082e10	1.22557	0.612786	+	0.790249i	$0.290049\pi$
$198$	0	0
$199$	−3.75183e10	−1.69592	−0.847959	−	0.530062i	$-0.822168\pi$
$199$	−3.75183e10	−1.69592	−0.847959	+	0.530062i	$0.822168\pi$
$200$	0	0
$201$	1.54591e10	0.668041
$202$	0	0
$203$	3.11548e9	0.128764
$204$	0	0
$205$	3.49110e9	0.138061
$206$	0	0
$207$	2.12245e10	0.803473
$208$	0	0
$209$	−2.93582e10	−1.06432
$210$	0	0
$211$	4.44170e9	0.154269	0.0771344	−	0.997021i	$-0.475423\pi$
$211$	4.44170e9	0.154269	0.0771344	+	0.997021i	$0.475423\pi$
$212$	0	0
$213$	−5.38073e10	−1.79115
$214$	0	0
$215$	−7.03291e7	−0.00224472
$216$	0	0
$217$	4.21650e9	0.129087
$218$	0	0
$219$	−1.15402e10	−0.339011
$220$	0	0
$221$	1.39753e10	0.394091
$222$	0	0
$223$	5.79794e10	1.57001	0.785003	−	0.619491i	$-0.212661\pi$
$223$	5.79794e10	1.57001	0.785003	+	0.619491i	$0.212661\pi$
$224$	0	0
$225$	−1.94455e10	−0.505823
$226$	0	0
$227$	−2.97908e10	−0.744674	−0.372337	−	0.928098i	$-0.621443\pi$
$227$	−2.97908e10	−0.744674	−0.372337	+	0.928098i	$0.621443\pi$
$228$	0	0
$229$	−5.33905e9	−0.128293	−0.0641467	−	0.997940i	$-0.520433\pi$
$229$	−5.33905e9	−0.128293	−0.0641467	+	0.997940i	$0.520433\pi$
$230$	0	0
$231$	−4.26489e9	−0.0985494
$232$	0	0
$233$	−6.03931e10	−1.34241	−0.671206	−	0.741271i	$-0.734223\pi$
$233$	−6.03931e10	−1.34241	−0.671206	+	0.741271i	$0.734223\pi$
$234$	0	0
$235$	−1.68823e9	−0.0361098
$236$	0	0
$237$	−5.13098e10	−1.05641
$238$	0	0
$239$	−2.37510e10	−0.470859	−0.235429	−	0.971891i	$-0.575650\pi$
$239$	−2.37510e10	−0.470859	−0.235429	+	0.971891i	$0.575650\pi$
$240$	0	0
$241$	−6.01204e10	−1.14801	−0.574005	−	0.818852i	$-0.694611\pi$
$241$	−6.01204e10	−1.14801	−0.574005	+	0.818852i	$0.694611\pi$
$242$	0	0
$243$	5.06581e10	0.932011
$244$	0	0
$245$	−4.29594e9	−0.0761748
$246$	0	0
$247$	2.08601e10	0.356600
$248$	0	0
$249$	−1.03610e11	−1.70807
$250$	0	0
$251$	6.69307e10	1.06437	0.532187	−	0.846627i	$-0.321370\pi$
$251$	6.69307e10	1.06437	0.532187	+	0.846627i	$0.321370\pi$
$252$	0	0
$253$	8.51840e10	1.30712
$254$	0	0
$255$	9.06212e9	0.134214
$256$	0	0
$257$	−2.21737e10	−0.317058	−0.158529	−	0.987354i	$-0.550675\pi$
$257$	−2.21737e10	−0.317058	−0.158529	+	0.987354i	$0.550675\pi$
$258$	0	0
$259$	−5.33624e9	−0.0736863
$260$	0	0
$261$	5.06797e10	0.676008
$262$	0	0
$263$	−1.17621e11	−1.51595	−0.757974	−	0.652284i	$-0.773811\pi$
$263$	−1.17621e11	−1.51595	−0.757974	+	0.652284i	$0.773811\pi$
$264$	0	0
$265$	2.68935e9	0.0334997
$266$	0	0
$267$	1.68372e11	2.02754
$268$	0	0
$269$	2.02220e10	0.235472	0.117736	−	0.993045i	$-0.462436\pi$
$269$	2.02220e10	0.235472	0.117736	+	0.993045i	$0.462436\pi$
$270$	0	0
$271$	7.34249e10	0.826955	0.413477	−	0.910514i	$-0.364314\pi$
$271$	7.34249e10	0.826955	0.413477	+	0.910514i	$0.364314\pi$
$272$	0	0
$273$	3.03037e9	0.0330190
$274$	0	0
$275$	−7.80440e10	−0.822891
$276$	0	0
$277$	1.34300e11	1.37062	0.685310	−	0.728252i	$-0.259667\pi$
$277$	1.34300e11	1.37062	0.685310	+	0.728252i	$0.259667\pi$
$278$	0	0
$279$	6.85900e10	0.677707
$280$	0	0
$281$	3.44336e10	0.329461	0.164730	−	0.986339i	$-0.447325\pi$
$281$	3.44336e10	0.329461	0.164730	+	0.986339i	$0.447325\pi$
$282$	0	0
$283$	1.61412e10	0.149588	0.0747939	−	0.997199i	$-0.476170\pi$
$283$	1.61412e10	0.149588	0.0747939	+	0.997199i	$0.476170\pi$
$284$	0	0
$285$	1.35265e10	0.121446
$286$	0	0
$287$	2.00006e10	0.174010
$288$	0	0
$289$	1.20842e11	1.01900
$290$	0	0
$291$	1.80338e11	1.47424
$292$	0	0
$293$	1.74179e10	0.138067	0.0690336	−	0.997614i	$-0.478008\pi$
$293$	1.74179e10	0.138067	0.0690336	+	0.997614i	$0.478008\pi$
$294$	0	0
$295$	3.24341e9	0.0249346
$296$	0	0
$297$	6.69689e10	0.499424
$298$	0	0
$299$	−6.05266e10	−0.437951
$300$	0	0
$301$	−4.02918e8	−0.00282922
$302$	0	0
$303$	−2.32654e11	−1.58569
$304$	0	0
$305$	−1.31760e9	−0.00871837
$306$	0	0
$307$	−1.82501e11	−1.17258	−0.586290	−	0.810101i	$-0.699412\pi$
$307$	−1.82501e11	−1.17258	−0.586290	+	0.810101i	$0.699412\pi$
$308$	0	0
$309$	8.46176e10	0.528017
$310$	0	0
$311$	2.24624e11	1.36155	0.680776	−	0.732492i	$-0.261643\pi$
$311$	2.24624e11	1.36155	0.680776	+	0.732492i	$0.261643\pi$
$312$	0	0
$313$	9.70945e10	0.571801	0.285901	−	0.958259i	$-0.407707\pi$
$313$	9.70945e10	0.571801	0.285901	+	0.958259i	$0.407707\pi$
$314$	0	0
$315$	6.62669e8	0.00379227
$316$	0	0
$317$	4.71097e10	0.262025	0.131013	−	0.991381i	$-0.458177\pi$
$317$	4.71097e10	0.262025	0.131013	+	0.991381i	$0.458177\pi$
$318$	0	0
$319$	2.03401e11	1.09976
$320$	0	0
$321$	4.32542e11	2.27382
$322$	0	0
$323$	3.57382e11	1.82693
$324$	0	0
$325$	5.54533e10	0.275710
$326$	0	0
$327$	3.48926e10	0.168760
$328$	0	0
$329$	−9.67190e9	−0.0455125
$330$	0	0
$331$	−2.08345e11	−0.954017	−0.477009	−	0.878899i	$-0.658279\pi$
$331$	−2.08345e11	−0.954017	−0.477009	+	0.878899i	$0.658279\pi$
$332$	0	0
$333$	−8.68049e10	−0.386852
$334$	0	0
$335$	−9.64039e9	−0.0418209
$336$	0	0
$337$	−3.96693e11	−1.67540	−0.837702	−	0.546128i	$-0.816101\pi$
$337$	−3.96693e11	−1.67540	−0.837702	+	0.546128i	$0.816101\pi$
$338$	0	0
$339$	3.04487e11	1.25219
$340$	0	0
$341$	2.75284e11	1.10252
$342$	0	0
$343$	−4.94566e10	−0.192930
$344$	0	0
$345$	−3.92477e10	−0.149152
$346$	0	0
$347$	−5.09902e11	−1.88801	−0.944005	−	0.329931i	$-0.892974\pi$
$347$	−5.09902e11	−1.88801	−0.944005	+	0.329931i	$0.892974\pi$
$348$	0	0
$349$	−5.23208e10	−0.188782	−0.0943908	−	0.995535i	$-0.530090\pi$
$349$	−5.23208e10	−0.188782	−0.0943908	+	0.995535i	$0.530090\pi$
$350$	0	0
$351$	−4.75840e10	−0.167332
$352$	0	0
$353$	4.58793e11	1.57264	0.786322	−	0.617817i	$-0.211983\pi$
$353$	4.58793e11	1.57264	0.786322	+	0.617817i	$0.211983\pi$
$354$	0	0
$355$	3.35545e10	0.112130
$356$	0	0
$357$	5.19172e10	0.169163
$358$	0	0
$359$	1.07780e11	0.342464	0.171232	−	0.985231i	$-0.445225\pi$
$359$	1.07780e11	0.342464	0.171232	+	0.985231i	$0.445225\pi$
$360$	0	0
$361$	2.10755e11	0.653123
$362$	0	0
$363$	1.27907e11	0.386646
$364$	0	0
$365$	7.19650e9	0.0212228
$366$	0	0
$367$	7.18381e10	0.206708	0.103354	−	0.994645i	$-0.467043\pi$
$367$	7.18381e10	0.206708	0.103354	+	0.994645i	$0.467043\pi$
$368$	0	0
$369$	3.25351e11	0.913552
$370$	0	0
$371$	1.54074e10	0.0422227
$372$	0	0
$373$	−6.20721e11	−1.66038	−0.830189	−	0.557482i	$-0.811767\pi$
$373$	−6.20721e11	−1.66038	−0.830189	+	0.557482i	$0.811767\pi$
$374$	0	0
$375$	7.21299e10	0.188354
$376$	0	0
$377$	−1.44525e11	−0.368473
$378$	0	0
$379$	3.52045e11	0.876438	0.438219	−	0.898868i	$-0.355609\pi$
$379$	3.52045e11	0.876438	0.438219	+	0.898868i	$0.355609\pi$
$380$	0	0
$381$	4.26617e11	1.03723
$382$	0	0
$383$	−8.06536e11	−1.91527	−0.957633	−	0.287991i	$-0.907013\pi$
$383$	−8.06536e11	−1.91527	−0.957633	+	0.287991i	$0.907013\pi$
$384$	0	0
$385$	2.65960e9	0.00616941
$386$	0	0
$387$	−6.55428e9	−0.0148534
$388$	0	0
$389$	5.94530e10	0.131644	0.0658219	−	0.997831i	$-0.479033\pi$
$389$	5.94530e10	0.131644	0.0658219	+	0.997831i	$0.479033\pi$
$390$	0	0
$391$	−1.03696e12	−2.24371
$392$	0	0
$393$	2.94353e11	0.622447
$394$	0	0
$395$	3.19970e10	0.0661337
$396$	0	0
$397$	−5.50704e11	−1.11266	−0.556328	−	0.830963i	$-0.687790\pi$
$397$	−5.50704e11	−1.11266	−0.556328	+	0.830963i	$0.687790\pi$
$398$	0	0
$399$	7.74937e10	0.153069
$400$	0	0
$401$	−7.13647e11	−1.37827	−0.689134	−	0.724634i	$-0.742009\pi$
$401$	−7.13647e11	−1.37827	−0.689134	+	0.724634i	$0.742009\pi$
$402$	0	0
$403$	−1.95600e11	−0.369400
$404$	0	0
$405$	−5.20404e10	−0.0961154
$406$	0	0
$407$	−3.48389e11	−0.629346
$408$	0	0
$409$	4.00062e11	0.706924	0.353462	−	0.935449i	$-0.385004\pi$
$409$	4.00062e11	0.706924	0.353462	+	0.935449i	$0.385004\pi$
$410$	0	0
$411$	1.81429e11	0.313630
$412$	0	0
$413$	1.85816e10	0.0314274
$414$	0	0
$415$	6.46117e10	0.106929
$416$	0	0
$417$	5.57612e10	0.0903066
$418$	0	0
$419$	2.27169e11	0.360070	0.180035	−	0.983660i	$-0.442379\pi$
$419$	2.27169e11	0.360070	0.180035	+	0.983660i	$0.442379\pi$
$420$	0	0
$421$	−9.80089e11	−1.52053	−0.760267	−	0.649611i	$-0.774932\pi$
$421$	−9.80089e11	−1.52053	−0.760267	+	0.649611i	$0.774932\pi$
$422$	0	0
$423$	−1.57333e11	−0.238940
$424$	0	0
$425$	9.50043e11	1.41251
$426$	0	0
$427$	−7.54859e9	−0.0109886
$428$	0	0
$429$	1.97845e11	0.282012
$430$	0	0
$431$	−4.10189e11	−0.572580	−0.286290	−	0.958143i	$-0.592422\pi$
$431$	−4.10189e11	−0.572580	−0.286290	+	0.958143i	$0.592422\pi$
$432$	0	0
$433$	2.86127e11	0.391168	0.195584	−	0.980687i	$-0.437340\pi$
$433$	2.86127e11	0.391168	0.195584	+	0.980687i	$0.437340\pi$
$434$	0	0
$435$	−9.37152e10	−0.125490
$436$	0	0
$437$	−1.54781e12	−2.03025
$438$	0	0
$439$	7.89980e11	1.01514	0.507570	−	0.861611i	$-0.330544\pi$
$439$	7.89980e11	1.01514	0.507570	+	0.861611i	$0.330544\pi$
$440$	0	0
$441$	−4.00358e11	−0.504052
$442$	0	0
$443$	1.25006e12	1.54210	0.771050	−	0.636774i	$-0.219732\pi$
$443$	1.25006e12	1.54210	0.771050	+	0.636774i	$0.219732\pi$
$444$	0	0
$445$	−1.04998e11	−0.126928
$446$	0	0
$447$	1.06934e12	1.26687
$448$	0	0
$449$	−1.34799e12	−1.56524	−0.782618	−	0.622503i	$-0.786116\pi$
$449$	−1.34799e12	−1.56524	−0.782618	+	0.622503i	$0.786116\pi$
$450$	0	0
$451$	1.30579e12	1.48620
$452$	0	0
$453$	8.07118e11	0.900524
$454$	0	0
$455$	−1.88975e9	−0.00206706
$456$	0	0
$457$	−1.45228e12	−1.55750	−0.778750	−	0.627334i	$-0.784146\pi$
$457$	−1.45228e12	−1.55750	−0.778750	+	0.627334i	$0.784146\pi$
$458$	0	0
$459$	−8.15224e11	−0.857274
$460$	0	0
$461$	−5.81071e9	−0.00599204	−0.00299602	−	0.999996i	$-0.500954\pi$
$461$	−5.81071e9	−0.00599204	−0.00299602	+	0.999996i	$0.500954\pi$
$462$	0	0
$463$	−7.63830e11	−0.772471	−0.386236	−	0.922400i	$-0.626225\pi$
$463$	−7.63830e11	−0.772471	−0.386236	+	0.922400i	$0.626225\pi$
$464$	0	0
$465$	−1.26834e11	−0.125805
$466$	0	0
$467$	5.64623e11	0.549329	0.274664	−	0.961540i	$-0.411433\pi$
$467$	5.64623e11	0.549329	0.274664	+	0.961540i	$0.411433\pi$
$468$	0	0
$469$	−5.52301e10	−0.0527106
$470$	0	0
$471$	−1.35686e12	−1.27040
$472$	0	0
$473$	−2.63054e10	−0.0241641
$474$	0	0
$475$	1.41807e12	1.27814
$476$	0	0
$477$	2.50633e11	0.221669
$478$	0	0
$479$	1.20508e12	1.04594	0.522970	−	0.852351i	$-0.324824\pi$
$479$	1.20508e12	1.04594	0.522970	+	0.852351i	$0.324824\pi$
$480$	0	0
$481$	2.47544e11	0.210863
$482$	0	0
$483$	−2.24851e11	−0.187989
$484$	0	0
$485$	−1.12460e11	−0.0922909
$486$	0	0
$487$	1.32287e12	1.06570	0.532852	−	0.846208i	$-0.321120\pi$
$487$	1.32287e12	1.06570	0.532852	+	0.846208i	$0.321120\pi$
$488$	0	0
$489$	1.45692e12	1.15225
$490$	0	0
$491$	−2.32552e11	−0.180573	−0.0902865	−	0.995916i	$-0.528778\pi$
$491$	−2.32552e11	−0.180573	−0.0902865	+	0.995916i	$0.528778\pi$
$492$	0	0
$493$	−2.47604e12	−1.88776
$494$	0	0
$495$	4.32639e10	0.0323893
$496$	0	0
$497$	1.92235e11	0.141328
$498$	0	0
$499$	−2.38399e12	−1.72128	−0.860641	−	0.509211i	$-0.829937\pi$
$499$	−2.38399e12	−1.72128	−0.860641	+	0.509211i	$0.829937\pi$
$500$	0	0
$501$	2.69731e11	0.191277
$502$	0	0
$503$	−7.19053e11	−0.500847	−0.250424	−	0.968136i	$-0.580570\pi$
$503$	−7.19053e11	−0.500847	−0.250424	+	0.968136i	$0.580570\pi$
$504$	0	0
$505$	1.45084e11	0.0992680
$506$	0	0
$507$	−1.40577e11	−0.0944881
$508$	0	0
$509$	−1.30257e12	−0.860146	−0.430073	−	0.902794i	$-0.641512\pi$
$509$	−1.30257e12	−0.860146	−0.430073	+	0.902794i	$0.641512\pi$
$510$	0	0
$511$	4.12290e10	0.0267491
$512$	0	0
$513$	−1.21684e12	−0.775717
$514$	0	0
$515$	−5.27679e10	−0.0330550
$516$	0	0
$517$	−6.31452e11	−0.388717
$518$	0	0
$519$	−3.68562e12	−2.22976
$520$	0	0
$521$	7.91136e11	0.470415	0.235208	−	0.971945i	$-0.424423\pi$
$521$	7.91136e11	0.470415	0.235208	+	0.971945i	$0.424423\pi$
$522$	0	0
$523$	1.57891e12	0.922783	0.461392	−	0.887197i	$-0.347350\pi$
$523$	1.57891e12	0.922783	0.461392	+	0.887197i	$0.347350\pi$
$524$	0	0
$525$	2.06005e11	0.118348
$526$	0	0
$527$	−3.35108e12	−1.89250
$528$	0	0
$529$	2.68987e12	1.49342
$530$	0	0
$531$	3.02268e11	0.164994
$532$	0	0
$533$	−9.27813e11	−0.497952
$534$	0	0
$535$	−2.69735e11	−0.142346
$536$	0	0
$537$	−2.78847e11	−0.144705
$538$	0	0
$539$	−1.60683e12	−0.820010
$540$	0	0
$541$	−7.58135e11	−0.380503	−0.190252	−	0.981735i	$-0.560930\pi$
$541$	−7.58135e11	−0.380503	−0.190252	+	0.981735i	$0.560930\pi$
$542$	0	0
$543$	1.20192e12	0.593302
$544$	0	0
$545$	−2.17592e10	−0.0105647
$546$	0	0
$547$	1.24661e12	0.595373	0.297687	−	0.954664i	$-0.403785\pi$
$547$	1.24661e12	0.595373	0.297687	+	0.954664i	$0.403785\pi$
$548$	0	0
$549$	−1.22793e11	−0.0576898
$550$	0	0
$551$	−3.69584e12	−1.70817
$552$	0	0
$553$	1.83312e11	0.0833543
$554$	0	0
$555$	1.60517e11	0.0718128
$556$	0	0
$557$	3.10874e12	1.36847	0.684236	−	0.729261i	$-0.260136\pi$
$557$	3.10874e12	1.36847	0.684236	+	0.729261i	$0.260136\pi$
$558$	0	0
$559$	1.86910e10	0.00809618
$560$	0	0
$561$	3.38953e12	1.44480
$562$	0	0
$563$	−3.78920e11	−0.158950	−0.0794749	−	0.996837i	$-0.525324\pi$
$563$	−3.78920e11	−0.158950	−0.0794749	+	0.996837i	$0.525324\pi$
$564$	0	0
$565$	−1.89879e11	−0.0783898
$566$	0	0
$567$	−2.98141e11	−0.121143
$568$	0	0
$569$	−2.66220e12	−1.06472	−0.532359	−	0.846518i	$-0.678695\pi$
$569$	−2.66220e12	−1.06472	−0.532359	+	0.846518i	$0.678695\pi$
$570$	0	0
$571$	−1.91714e12	−0.754730	−0.377365	−	0.926065i	$-0.623170\pi$
$571$	−1.91714e12	−0.754730	−0.377365	+	0.926065i	$0.623170\pi$
$572$	0	0
$573$	3.93561e12	1.52516
$574$	0	0
$575$	−4.11459e12	−1.56972
$576$	0	0
$577$	−3.21629e12	−1.20799	−0.603996	−	0.796987i	$-0.706426\pi$
$577$	−3.21629e12	−1.20799	−0.603996	+	0.796987i	$0.706426\pi$
$578$	0	0
$579$	4.81455e12	1.78033
$580$	0	0
$581$	3.70163e11	0.134772
$582$	0	0
$583$	1.00591e12	0.360620
$584$	0	0
$585$	−3.07407e10	−0.0108521
$586$	0	0
$587$	6.43183e11	0.223595	0.111798	−	0.993731i	$-0.464339\pi$
$587$	6.43183e11	0.223595	0.111798	+	0.993731i	$0.464339\pi$
$588$	0	0
$589$	−5.00195e12	−1.71246
$590$	0	0
$591$	−4.46481e12	−1.50542
$592$	0	0
$593$	−3.55490e12	−1.18054	−0.590270	−	0.807206i	$-0.700979\pi$
$593$	−3.55490e12	−1.18054	−0.590270	+	0.807206i	$0.700979\pi$
$594$	0	0
$595$	−3.23758e10	−0.0105900
$596$	0	0
$597$	6.46561e12	2.08317
$598$	0	0
$599$	1.12210e12	0.356131	0.178066	−	0.984019i	$-0.443016\pi$
$599$	1.12210e12	0.356131	0.178066	+	0.984019i	$0.443016\pi$
$600$	0	0
$601$	−5.71984e12	−1.78834	−0.894168	−	0.447732i	$-0.852232\pi$
$601$	−5.71984e12	−1.78834	−0.894168	+	0.447732i	$0.852232\pi$
$602$	0	0
$603$	−8.98431e11	−0.276730
$604$	0	0
$605$	−7.97632e10	−0.0242049
$606$	0	0
$607$	−5.37924e11	−0.160832	−0.0804158	−	0.996761i	$-0.525625\pi$
$607$	−5.37924e11	−0.160832	−0.0804158	+	0.996761i	$0.525625\pi$
$608$	0	0
$609$	−5.36897e11	−0.158166
$610$	0	0
$611$	4.48672e11	0.130240
$612$	0	0
$613$	−9.62972e11	−0.275449	−0.137725	−	0.990471i	$-0.543979\pi$
$613$	−9.62972e11	−0.275449	−0.137725	+	0.990471i	$0.543979\pi$
$614$	0	0
$615$	−6.01628e11	−0.169586
$616$	0	0
$617$	−2.70351e12	−0.751008	−0.375504	−	0.926821i	$-0.622530\pi$
$617$	−2.70351e12	−0.751008	−0.375504	+	0.926821i	$0.622530\pi$
$618$	0	0
$619$	−2.10442e11	−0.0576136	−0.0288068	−	0.999585i	$-0.509171\pi$
$619$	−2.10442e11	−0.0576136	−0.0288068	+	0.999585i	$0.509171\pi$
$620$	0	0
$621$	3.53070e12	0.952683
$622$	0	0
$623$	−6.01534e11	−0.159979
$624$	0	0
$625$	3.74716e12	0.982295
$626$	0	0
$627$	5.05935e12	1.30735
$628$	0	0
$629$	4.24100e12	1.08029
$630$	0	0
$631$	−3.07558e12	−0.772315	−0.386157	−	0.922433i	$-0.626198\pi$
$631$	−3.07558e12	−0.772315	−0.386157	+	0.922433i	$0.626198\pi$
$632$	0	0
$633$	−7.65447e11	−0.189495
$634$	0	0
$635$	−2.66040e11	−0.0649330
$636$	0	0
$637$	1.14171e12	0.274745
$638$	0	0
$639$	3.12709e12	0.741970
$640$	0	0
$641$	5.94034e12	1.38979	0.694897	−	0.719110i	$-0.255450\pi$
$641$	5.94034e12	1.38979	0.694897	+	0.719110i	$0.255450\pi$
$642$	0	0
$643$	5.38874e12	1.24319	0.621595	−	0.783339i	$-0.286485\pi$
$643$	5.38874e12	1.24319	0.621595	+	0.783339i	$0.286485\pi$
$644$	0	0
$645$	1.21200e10	0.00275729
$646$	0	0
$647$	1.71794e12	0.385423	0.192712	−	0.981255i	$-0.438272\pi$
$647$	1.71794e12	0.385423	0.192712	+	0.981255i	$0.438272\pi$
$648$	0	0
$649$	1.21314e12	0.268418
$650$	0	0
$651$	−7.26638e11	−0.158564
$652$	0	0
$653$	4.25971e12	0.916793	0.458396	−	0.888748i	$-0.348424\pi$
$653$	4.25971e12	0.916793	0.458396	+	0.888748i	$0.348424\pi$
$654$	0	0
$655$	−1.83560e11	−0.0389665
$656$	0	0
$657$	6.70674e11	0.140432
$658$	0	0
$659$	6.15003e12	1.27026	0.635130	−	0.772405i	$-0.280946\pi$
$659$	6.15003e12	1.27026	0.635130	+	0.772405i	$0.280946\pi$
$660$	0	0
$661$	7.53625e12	1.53550	0.767748	−	0.640752i	$-0.221377\pi$
$661$	7.53625e12	1.53550	0.767748	+	0.640752i	$0.221377\pi$
$662$	0	0
$663$	−2.40840e12	−0.484080
$664$	0	0
$665$	−4.83254e10	−0.00958248
$666$	0	0
$667$	1.07236e13	2.09785
$668$	0	0
$669$	−9.99170e12	−1.92851
$670$	0	0
$671$	−4.92827e11	−0.0938520
$672$	0	0
$673$	1.99916e12	0.375646	0.187823	−	0.982203i	$-0.439857\pi$
$673$	1.99916e12	0.375646	0.187823	+	0.982203i	$0.439857\pi$
$674$	0	0
$675$	−3.23476e12	−0.599757
$676$	0	0
$677$	5.28289e11	0.0966546	0.0483273	−	0.998832i	$-0.484611\pi$
$677$	5.28289e11	0.0966546	0.0483273	+	0.998832i	$0.484611\pi$
$678$	0	0
$679$	−6.44285e11	−0.116323
$680$	0	0
$681$	5.13391e12	0.914717
$682$	0	0
$683$	−6.54886e12	−1.15152	−0.575761	−	0.817618i	$-0.695294\pi$
$683$	−6.54886e12	−1.15152	−0.575761	+	0.817618i	$0.695294\pi$
$684$	0	0
$685$	−1.13140e11	−0.0196339
$686$	0	0
$687$	9.20090e11	0.157589
$688$	0	0
$689$	−7.14736e11	−0.120826
$690$	0	0
$691$	4.45526e12	0.743400	0.371700	−	0.928353i	$-0.378775\pi$
$691$	4.45526e12	0.743400	0.371700	+	0.928353i	$0.378775\pi$
$692$	0	0
$693$	2.47860e11	0.0408232
$694$	0	0
$695$	−3.47729e10	−0.00565340
$696$	0	0
$697$	−1.58956e13	−2.55110
$698$	0	0
$699$	1.04077e13	1.64894
$700$	0	0
$701$	7.20107e12	1.12633	0.563166	−	0.826344i	$-0.309583\pi$
$701$	7.20107e12	1.12633	0.563166	+	0.826344i	$0.309583\pi$
$702$	0	0
$703$	6.33028e12	0.977516
$704$	0	0
$705$	2.90935e11	0.0443553
$706$	0	0
$707$	8.31193e11	0.125116
$708$	0	0
$709$	5.87074e12	0.872538	0.436269	−	0.899816i	$-0.356300\pi$
$709$	5.87074e12	0.872538	0.436269	+	0.899816i	$0.356300\pi$
$710$	0	0
$711$	2.98194e12	0.437609
$712$	0	0
$713$	1.45134e13	2.10313
$714$	0	0
$715$	−1.23377e11	−0.0176546
$716$	0	0
$717$	4.09305e12	0.578377
$718$	0	0
$719$	7.39696e12	1.03222	0.516111	−	0.856522i	$-0.327379\pi$
$719$	7.39696e12	1.03222	0.516111	+	0.856522i	$0.327379\pi$
$720$	0	0
$721$	−3.02309e11	−0.0416622
$722$	0	0
$723$	1.03607e13	1.41015
$724$	0	0
$725$	−9.82479e12	−1.32069
$726$	0	0
$727$	−1.15600e13	−1.53481	−0.767404	−	0.641163i	$-0.778452\pi$
$727$	−1.15600e13	−1.53481	−0.767404	+	0.641163i	$0.778452\pi$
$728$	0	0
$729$	8.01383e11	0.105091
$730$	0	0
$731$	3.20220e11	0.0414783
$732$	0	0
$733$	2.14718e12	0.274726	0.137363	−	0.990521i	$-0.456137\pi$
$733$	2.14718e12	0.274726	0.137363	+	0.990521i	$0.456137\pi$
$734$	0	0
$735$	7.40329e11	0.0935689
$736$	0	0
$737$	−3.60582e12	−0.450195
$738$	0	0
$739$	1.24299e13	1.53310	0.766548	−	0.642187i	$-0.221973\pi$
$739$	1.24299e13	1.53310	0.766548	+	0.642187i	$0.221973\pi$
$740$	0	0
$741$	−3.59487e12	−0.438027
$742$	0	0
$743$	−7.19308e12	−0.865896	−0.432948	−	0.901419i	$-0.642527\pi$
$743$	−7.19308e12	−0.865896	−0.432948	+	0.901419i	$0.642527\pi$
$744$	0	0
$745$	−6.66844e11	−0.0793087
$746$	0	0
$747$	6.02145e12	0.707552
$748$	0	0
$749$	−1.54532e12	−0.179411
$750$	0	0
$751$	1.64354e13	1.88539	0.942694	−	0.333660i	$-0.108284\pi$
$751$	1.64354e13	1.88539	0.942694	+	0.333660i	$0.108284\pi$
$752$	0	0
$753$	−1.15343e13	−1.30742
$754$	0	0
$755$	−5.03323e11	−0.0563748
$756$	0	0
$757$	−1.72629e13	−1.91066	−0.955329	−	0.295545i	$-0.904499\pi$
$757$	−1.72629e13	−1.91066	−0.955329	+	0.295545i	$0.904499\pi$
$758$	0	0
$759$	−1.46799e13	−1.60559
$760$	0	0
$761$	−7.28511e12	−0.787417	−0.393709	−	0.919235i	$-0.628808\pi$
$761$	−7.28511e12	−0.787417	−0.393709	+	0.919235i	$0.628808\pi$
$762$	0	0
$763$	−1.24659e11	−0.0133157
$764$	0	0
$765$	−5.26658e11	−0.0555972
$766$	0	0
$767$	−8.61986e11	−0.0899335
$768$	0	0
$769$	−6.42531e12	−0.662561	−0.331280	−	0.943532i	$-0.607481\pi$
$769$	−6.42531e12	−0.662561	−0.331280	+	0.943532i	$0.607481\pi$
$770$	0	0
$771$	3.82124e12	0.389457
$772$	0	0
$773$	−1.16581e12	−0.117441	−0.0587205	−	0.998274i	$-0.518702\pi$
$773$	−1.16581e12	−0.117441	−0.0587205	+	0.998274i	$0.518702\pi$
$774$	0	0
$775$	−1.32969e13	−1.32401
$776$	0	0
$777$	9.19606e11	0.0905122
$778$	0	0
$779$	−2.37263e13	−2.30841
$780$	0	0
$781$	1.25505e13	1.20706
$782$	0	0
$783$	8.43057e12	0.801547
$784$	0	0
$785$	8.46141e11	0.0795297
$786$	0	0
$787$	1.34999e13	1.25442	0.627212	−	0.778849i	$-0.284196\pi$
$787$	1.34999e13	1.25442	0.627212	+	0.778849i	$0.284196\pi$
$788$	0	0
$789$	2.02699e13	1.86211
$790$	0	0
$791$	−1.08782e12	−0.0988017
$792$	0	0
$793$	3.50173e11	0.0314451
$794$	0	0
$795$	−4.63462e11	−0.0411492
$796$	0	0
$797$	−5.19207e12	−0.455804	−0.227902	−	0.973684i	$-0.573187\pi$
$797$	−5.19207e12	−0.455804	−0.227902	+	0.973684i	$0.573187\pi$
$798$	0	0
$799$	7.68678e12	0.667242
$800$	0	0
$801$	−9.78519e12	−0.839891
$802$	0	0
$803$	2.69173e12	0.228461
$804$	0	0
$805$	1.40218e11	0.0117685
$806$	0	0
$807$	−3.48490e12	−0.289241
$808$	0	0
$809$	1.10307e13	0.905392	0.452696	−	0.891665i	$-0.350462\pi$
$809$	1.10307e13	0.905392	0.452696	+	0.891665i	$0.350462\pi$
$810$	0	0
$811$	−8.42266e12	−0.683684	−0.341842	−	0.939757i	$-0.611051\pi$
$811$	−8.42266e12	−0.683684	−0.341842	+	0.939757i	$0.611051\pi$
$812$	0	0
$813$	−1.26535e13	−1.01579
$814$	0	0
$815$	−9.08541e11	−0.0721332
$816$	0	0
$817$	4.77974e11	0.0375322
$818$	0	0
$819$	−1.76114e11	−0.0136778
$820$	0	0
$821$	−1.08676e13	−0.834815	−0.417408	−	0.908719i	$-0.637061\pi$
$821$	−1.08676e13	−0.834815	−0.417408	+	0.908719i	$0.637061\pi$
$822$	0	0
$823$	−2.46796e13	−1.87516	−0.937579	−	0.347771i	$-0.886939\pi$
$823$	−2.46796e13	−1.87516	−0.937579	+	0.347771i	$0.886939\pi$
$824$	0	0
$825$	1.34495e13	1.01079
$826$	0	0
$827$	−4.08610e12	−0.303763	−0.151881	−	0.988399i	$-0.548533\pi$
$827$	−4.08610e12	−0.303763	−0.151881	+	0.988399i	$0.548533\pi$
$828$	0	0
$829$	−2.36893e12	−0.174203	−0.0871016	−	0.996199i	$-0.527760\pi$
$829$	−2.36893e12	−0.174203	−0.0871016	+	0.996199i	$0.527760\pi$
$830$	0	0
$831$	−2.31442e13	−1.68359
$832$	0	0
$833$	1.95602e13	1.40757
$834$	0	0
$835$	−1.68206e11	−0.0119743
$836$	0	0
$837$	1.14100e13	0.803562
$838$	0	0
$839$	−7.06718e12	−0.492399	−0.246200	−	0.969219i	$-0.579182\pi$
$839$	−7.06718e12	−0.492399	−0.246200	+	0.969219i	$0.579182\pi$
$840$	0	0
$841$	1.10986e13	0.765045
$842$	0	0
$843$	−5.93400e12	−0.404691
$844$	0	0
$845$	8.76641e10	0.00591516
$846$	0	0
$847$	−4.56966e11	−0.0305076
$848$	0	0
$849$	−2.78164e12	−0.183745
$850$	0	0
$851$	−1.83676e13	−1.20052
$852$	0	0
$853$	−1.40710e13	−0.910026	−0.455013	−	0.890485i	$-0.650365\pi$
$853$	−1.40710e13	−0.910026	−0.455013	+	0.890485i	$0.650365\pi$
$854$	0	0
$855$	−7.86111e11	−0.0503079
$856$	0	0
$857$	6.14374e12	0.389062	0.194531	−	0.980896i	$-0.437681\pi$
$857$	6.14374e12	0.389062	0.194531	+	0.980896i	$0.437681\pi$
$858$	0	0
$859$	1.52610e13	0.956346	0.478173	−	0.878266i	$-0.341299\pi$
$859$	1.52610e13	0.956346	0.478173	+	0.878266i	$0.341299\pi$
$860$	0	0
$861$	−3.44675e12	−0.213745
$862$	0	0
$863$	−9.38994e12	−0.576255	−0.288127	−	0.957592i	$-0.593033\pi$
$863$	−9.38994e12	−0.576255	−0.288127	+	0.957592i	$0.593033\pi$
$864$	0	0
$865$	2.29837e12	0.139588
$866$	0	0
$867$	−2.08249e13	−1.25169
$868$	0	0
$869$	1.19679e13	0.711919
$870$	0	0
$871$	2.56208e12	0.150838
$872$	0	0
$873$	−1.04806e13	−0.610692
$874$	0	0
$875$	−2.57695e11	−0.0148617
$876$	0	0
$877$	−2.11533e13	−1.20748	−0.603739	−	0.797182i	$-0.706323\pi$
$877$	−2.11533e13	−1.20748	−0.603739	+	0.797182i	$0.706323\pi$
$878$	0	0
$879$	−3.00165e12	−0.169594
$880$	0	0
$881$	−1.72011e13	−0.961975	−0.480988	−	0.876727i	$-0.659722\pi$
$881$	−1.72011e13	−0.961975	−0.480988	+	0.876727i	$0.659722\pi$
$882$	0	0
$883$	−1.16188e12	−0.0643188	−0.0321594	−	0.999483i	$-0.510238\pi$
$883$	−1.16188e12	−0.0643188	−0.0321594	+	0.999483i	$0.510238\pi$
$884$	0	0
$885$	−5.58944e11	−0.0306283
$886$	0	0
$887$	−1.29934e13	−0.704802	−0.352401	−	0.935849i	$-0.614635\pi$
$887$	−1.29934e13	−0.704802	−0.352401	+	0.935849i	$0.614635\pi$
$888$	0	0
$889$	−1.52415e12	−0.0818409
$890$	0	0
$891$	−1.94648e13	−1.03467
$892$	0	0
$893$	1.14736e13	0.603765
$894$	0	0
$895$	1.73890e11	0.00905883
$896$	0	0
$897$	1.04307e13	0.537955
$898$	0	0
$899$	3.46549e13	1.76948
$900$	0	0
$901$	−1.22451e13	−0.619013
$902$	0	0
$903$	6.94357e10	0.00347526
$904$	0	0
$905$	−7.49522e11	−0.0371420
$906$	0	0
$907$	1.30249e13	0.639059	0.319530	−	0.947576i	$-0.396475\pi$
$907$	1.30249e13	0.639059	0.319530	+	0.947576i	$0.396475\pi$
$908$	0	0
$909$	1.35210e13	0.656860
$910$	0	0
$911$	−3.46653e13	−1.66749	−0.833744	−	0.552151i	$-0.813807\pi$
$911$	−3.46653e13	−1.66749	−0.833744	+	0.552151i	$0.813807\pi$
$912$	0	0
$913$	2.41669e13	1.15107
$914$	0	0
$915$	2.27065e11	0.0107092
$916$	0	0
$917$	−1.05162e12	−0.0491131
$918$	0	0
$919$	2.36645e13	1.09440	0.547201	−	0.837001i	$-0.315693\pi$
$919$	2.36645e13	1.09440	0.547201	+	0.837001i	$0.315693\pi$
$920$	0	0
$921$	3.14507e13	1.44033
$922$	0	0
$923$	−8.91761e12	−0.404428
$924$	0	0
$925$	1.68280e13	0.755781
$926$	0	0
$927$	−4.91768e12	−0.218726
$928$	0	0
$929$	−1.42085e13	−0.625860	−0.312930	−	0.949776i	$-0.601311\pi$
$929$	−1.42085e13	−0.625860	−0.312930	+	0.949776i	$0.601311\pi$
$930$	0	0
$931$	2.91963e13	1.27366
$932$	0	0
$933$	−3.87099e13	−1.67246
$934$	0	0
$935$	−2.11373e12	−0.0904476
$936$	0	0
$937$	−3.78343e13	−1.60346	−0.801729	−	0.597688i	$-0.796086\pi$
$937$	−3.78343e13	−1.60346	−0.801729	+	0.597688i	$0.796086\pi$
$938$	0	0
$939$	−1.67325e13	−0.702369
$940$	0	0
$941$	−1.21944e13	−0.507000	−0.253500	−	0.967335i	$-0.581582\pi$
$941$	−1.21944e13	−0.507000	−0.253500	+	0.967335i	$0.581582\pi$
$942$	0	0
$943$	6.88430e13	2.83502
$944$	0	0
$945$	1.10235e11	0.00449652
$946$	0	0
$947$	4.76626e13	1.92576	0.962882	−	0.269922i	$-0.0869979\pi$
$947$	4.76626e13	1.92576	0.962882	+	0.269922i	$0.0869979\pi$
$948$	0	0
$949$	−1.91258e12	−0.0765459
$950$	0	0
$951$	−8.11850e12	−0.321857
$952$	0	0
$953$	3.32853e13	1.30718	0.653589	−	0.756850i	$-0.273263\pi$
$953$	3.32853e13	1.30718	0.653589	+	0.756850i	$0.273263\pi$
$954$	0	0
$955$	−2.45427e12	−0.0954787
$956$	0	0
$957$	−3.50526e13	−1.35088
$958$	0	0
$959$	−6.48181e11	−0.0247464
$960$	0	0
$961$	2.04624e13	0.773930
$962$	0	0
$963$	−2.51378e13	−0.941908
$964$	0	0
$965$	−3.00237e12	−0.111453
$966$	0	0
$967$	−3.78192e13	−1.39089	−0.695445	−	0.718579i	$-0.744793\pi$
$967$	−3.78192e13	−1.39089	−0.695445	+	0.718579i	$0.744793\pi$
$968$	0	0
$969$	−6.15883e13	−2.24410
$970$	0	0
$971$	−3.76386e12	−0.135877	−0.0679387	−	0.997689i	$-0.521642\pi$
$971$	−3.76386e12	−0.135877	−0.0679387	+	0.997689i	$0.521642\pi$
$972$	0	0
$973$	−1.99215e11	−0.00712549
$974$	0	0
$975$	−9.55639e12	−0.338667
$976$	0	0
$977$	−1.30991e13	−0.459956	−0.229978	−	0.973196i	$-0.573865\pi$
$977$	−1.30991e13	−0.459956	−0.229978	+	0.973196i	$0.573865\pi$
$978$	0	0
$979$	−3.92726e13	−1.36637
$980$	0	0
$981$	−2.02783e12	−0.0699072
$982$	0	0
$983$	3.20632e13	1.09526	0.547628	−	0.836722i	$-0.315531\pi$
$983$	3.20632e13	1.09526	0.547628	+	0.836722i	$0.315531\pi$
$984$	0	0
$985$	2.78427e12	0.0942429
$986$	0	0
$987$	1.66678e12	0.0559050
$988$	0	0
$989$	−1.38686e12	−0.0460945
$990$	0	0
$991$	−1.21061e13	−0.398725	−0.199363	−	0.979926i	$-0.563887\pi$
$991$	−1.21061e13	−0.398725	−0.199363	+	0.979926i	$0.563887\pi$
$992$	0	0
$993$	3.59044e13	1.17186
$994$	0	0
$995$	−4.03198e12	−0.130411
$996$	0	0
$997$	2.24835e13	0.720669	0.360335	−	0.932823i	$-0.382663\pi$
$997$	2.24835e13	0.720669	0.360335	+	0.932823i	$0.382663\pi$
$998$	0	0
$999$	−1.44400e13	−0.458693

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	208.10.a.j.1.1		6
4.3	odd	2		104.10.a.b.1.6	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
104.10.a.b.1.6	✓	6	4.3	odd	2
208.10.a.j.1.1		6	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 $\F_{q}$

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

\(f(q)\)	\(=\)	\(q-172.332 q^{3} +107.467 q^{5} +615.682 q^{7} +10015.3 q^{9} +40196.2 q^{11} -28561.0 q^{13} -18520.0 q^{15} -489315. q^{17} -730371. q^{19} -106102. q^{21} +2.11920e6 q^{23} -1.94158e6 q^{25} +1.66605e6 q^{27} +5.06021e6 q^{29} +6.84851e6 q^{31} -6.92710e6 q^{33} +66165.5 q^{35} -8.66721e6 q^{37} +4.92197e6 q^{39} +3.24853e7 q^{41} -654425. q^{43} +1.07632e6 q^{45} -1.57092e7 q^{47} -3.99745e7 q^{49} +8.43247e7 q^{51} +2.50249e7 q^{53} +4.31977e6 q^{55} +1.25866e8 q^{57} +3.01805e7 q^{59} -1.22605e7 q^{61} +6.16626e6 q^{63} -3.06937e6 q^{65} -8.97056e7 q^{67} -3.65207e8 q^{69} +3.12230e8 q^{71} +6.69648e7 q^{73} +3.34596e8 q^{75} +2.47481e7 q^{77} +2.97738e8 q^{79} -4.84245e8 q^{81} +6.01224e8 q^{83} -5.25853e7 q^{85} -8.72037e8 q^{87} -9.77021e8 q^{89} -1.75845e7 q^{91} -1.18022e9 q^{93} -7.84908e7 q^{95} -1.04646e9 q^{97} +4.02578e8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 60 q^{3} + 176 q^{5} - 3416 q^{7} + 22010 q^{9} + 101316 q^{11} - 171366 q^{13} - 257460 q^{15} - 170304 q^{17} + 92084 q^{19} - 424004 q^{21} - 2369944 q^{23} + 2043598 q^{25} + 6428196 q^{27} - 9021404 q^{29}+ \cdots - 282095980 q^{99}+O(q^{100}) \) 6 * q - 60 * q^3 + 176 * q^5 - 3416 * q^7 + 22010 * q^9 + 101316 * q^11 - 171366 * q^13 - 257460 * q^15 - 170304 * q^17 + 92084 * q^19 - 424004 * q^21 - 2369944 * q^23 + 2043598 * q^25 + 6428196 * q^27 - 9021404 * q^29 + 9894388 * q^31 + 6717200 * q^33 + 1910916 * q^35 + 2974376 * q^37 + 1713660 * q^39 - 2977668 * q^41 + 31585340 * q^43 - 99427116 * q^45 + 1766688 * q^47 - 25397750 * q^49 + 68102932 * q^51 - 137836444 * q^53 + 64702520 * q^55 - 253569848 * q^57 + 128523972 * q^59 + 33822404 * q^61 + 359085076 * q^63 - 5026736 * q^65 + 618333476 * q^67 - 837582264 * q^69 + 52534456 * q^71 - 667586460 * q^73 + 940557360 * q^75 - 944797672 * q^77 + 399417496 * q^79 - 821518498 * q^81 + 467205356 * q^83 - 2583200620 * q^85 + 219585328 * q^87 - 1630231268 * q^89 + 97564376 * q^91 - 585888632 * q^93 + 3079232768 * q^95 - 3266492188 * q^97 - 282095980 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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