LMFDB - Embedded newform 208.10.a.m.1.7

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:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 3 x^{7} - 135356 x^{6} - 24398 x^{5} + 5213582205 x^{4} + 598076469 x^{3} + \cdots + 15\!\cdots\!64 $ x^8 - 3x^7 - 135356x^6 - 24398x^5 + 5213582205x^4 + 598076469x^3 - 44131402568826x^2 + 60107523849756*x + 1558753465556664
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{25}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 104)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$227.704$ 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)

$f(q)$	$=$	$q+209.704 q^{3} +488.996 q^{5} -11010.2 q^{7} +24292.6 q^{9} -23581.1 q^{11} +28561.0 q^{13} +102544. q^{15} +160701. q^{17} +397828. q^{19} -2.30887e6 q^{21} -324898. q^{23} -1.71401e6 q^{25} +966648. q^{27} +4.55158e6 q^{29} +9.68113e6 q^{31} -4.94504e6 q^{33} -5.38392e6 q^{35} +5.70767e6 q^{37} +5.98934e6 q^{39} +1.10096e7 q^{41} -2.05034e7 q^{43} +1.18790e7 q^{45} +4.62183e7 q^{47} +8.08698e7 q^{49} +3.36996e7 q^{51} +9.51270e7 q^{53} -1.15311e7 q^{55} +8.34260e7 q^{57} -2.01368e7 q^{59} +1.10626e8 q^{61} -2.67465e8 q^{63} +1.39662e7 q^{65} +1.26183e8 q^{67} -6.81323e7 q^{69} +1.44850e8 q^{71} -3.98283e7 q^{73} -3.59434e8 q^{75} +2.59632e8 q^{77} -2.09504e8 q^{79} -2.75441e8 q^{81} -1.27715e8 q^{83} +7.85822e7 q^{85} +9.54482e8 q^{87} -1.03596e9 q^{89} -3.14461e8 q^{91} +2.03017e9 q^{93} +1.94536e8 q^{95} +1.07626e9 q^{97} -5.72846e8 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 141 q^{3} + 2051 q^{5} + 2417 q^{7} + 115741 q^{9} + 53118 q^{11} + 228488 q^{13} + 464555 q^{15} + 433095 q^{17} + 434954 q^{19} + 906875 q^{21} + 1124296 q^{23} + 5966065 q^{25} - 7820643 q^{27}+ \cdots + 641626736 q^{99}+O(q^{100}) $ 8 * q - 141 * q^3 + 2051 * q^5 + 2417 * q^7 + 115741 * q^9 + 53118 * q^11 + 228488 * q^13 + 464555 * q^15 + 433095 * q^17 + 434954 * q^19 + 906875 * q^21 + 1124296 * q^23 + 5966065 * q^25 - 7820643 * q^27 - 1913432 * q^29 - 5220148 * q^31 - 1626882 * q^33 - 950707 * q^35 - 419409 * q^37 - 4027101 * q^39 + 12148046 * q^41 - 32143187 * q^43 + 18893544 * q^45 - 57898411 * q^47 + 104302637 * q^49 - 198521477 * q^51 + 273106866 * q^53 - 141214662 * q^55 + 253032058 * q^57 - 50734502 * q^59 + 139520018 * q^61 + 20344158 * q^63 + 58578611 * q^65 - 528144786 * q^67 + 483506576 * q^69 - 334814973 * q^71 + 630864344 * q^73 - 2265796264 * q^75 + 939664146 * q^77 - 865037608 * q^79 + 3194281040 * q^81 - 1520229952 * q^83 + 2130158385 * q^85 - 1167867006 * q^87 + 894700240 * q^89 + 69031937 * q^91 + 3803209548 * q^93 + 2695926542 * q^95 + 980232164 * q^97 + 641626736 * 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
$3$	209.704	1.49472	0.747361	−	0.664419i	$-0.231321\pi$
$3$	209.704	1.49472	0.747361	+	0.664419i	$0.231321\pi$
$4$	0	0
$5$	488.996	0.349897	0.174949	−	0.984578i	$-0.444024\pi$
$5$	488.996	0.349897	0.174949	+	0.984578i	$0.444024\pi$
$6$	0	0
$7$	−11010.2	−1.73321	−0.866607	−	0.498992i	$-0.833704\pi$
$7$	−11010.2	−1.73321	−0.866607	+	0.498992i	$0.833704\pi$
$8$	0	0
$9$	24292.6	1.23419
$10$	0	0
$11$	−23581.1	−0.485621	−0.242810	−	0.970074i	$-0.578069\pi$
$11$	−23581.1	−0.485621	−0.242810	+	0.970074i	$0.578069\pi$
$12$	0	0
$13$	28561.0	0.277350
$13$	28561.0	0.277350
$14$	0	0
$15$	102544.	0.522999
$16$	0	0
$17$	160701.	0.466658	0.233329	−	0.972398i	$-0.425038\pi$
$17$	160701.	0.466658	0.233329	+	0.972398i	$0.425038\pi$
$18$	0	0
$19$	397828.	0.700332	0.350166	−	0.936688i	$-0.386125\pi$
$19$	397828.	0.700332	0.350166	+	0.936688i	$0.386125\pi$
$20$	0	0
$21$	−2.30887e6	−2.59067
$22$	0	0
$23$	−324898.	−0.242087	−0.121044	−	0.992647i	$-0.538624\pi$
$23$	−324898.	−0.242087	−0.121044	+	0.992647i	$0.538624\pi$
$24$	0	0
$25$	−1.71401e6	−0.877572
$26$	0	0
$27$	966648.	0.350051
$28$	0	0
$29$	4.55158e6	1.19501	0.597504	−	0.801866i	$-0.296159\pi$
$29$	4.55158e6	1.19501	0.597504	+	0.801866i	$0.296159\pi$
$30$	0	0
$31$	9.68113e6	1.88278	0.941388	−	0.337326i	$-0.109523\pi$
$31$	9.68113e6	1.88278	0.941388	+	0.337326i	$0.109523\pi$
$32$	0	0
$33$	−4.94504e6	−0.725867
$34$	0	0
$35$	−5.38392e6	−0.606447
$36$	0	0
$37$	5.70767e6	0.500670	0.250335	−	0.968159i	$-0.419459\pi$
$37$	5.70767e6	0.500670	0.250335	+	0.968159i	$0.419459\pi$
$38$	0	0
$39$	5.98934e6	0.414561
$40$	0	0
$41$	1.10096e7	0.608477	0.304239	−	0.952596i	$-0.401598\pi$
$41$	1.10096e7	0.608477	0.304239	+	0.952596i	$0.401598\pi$
$42$	0	0
$43$	−2.05034e7	−0.914574	−0.457287	−	0.889319i	$-0.651179\pi$
$43$	−2.05034e7	−0.914574	−0.457287	+	0.889319i	$0.651179\pi$
$44$	0	0
$45$	1.18790e7	0.431840
$46$	0	0
$47$	4.62183e7	1.38157	0.690787	−	0.723058i	$-0.257264\pi$
$47$	4.62183e7	1.38157	0.690787	+	0.723058i	$0.257264\pi$
$48$	0	0
$49$	8.08698e7	2.00403
$50$	0	0
$51$	3.36996e7	0.697524
$52$	0	0
$53$	9.51270e7	1.65601	0.828004	−	0.560722i	$-0.189476\pi$
$53$	9.51270e7	1.65601	0.828004	+	0.560722i	$0.189476\pi$
$54$	0	0
$55$	−1.15311e7	−0.169917
$56$	0	0
$57$	8.34260e7	1.04680
$58$	0	0
$59$	−2.01368e7	−0.216350	−0.108175	−	0.994132i	$-0.534501\pi$
$59$	−2.01368e7	−0.216350	−0.108175	+	0.994132i	$0.534501\pi$
$60$	0	0
$61$	1.10626e8	1.02299	0.511495	−	0.859286i	$-0.329092\pi$
$61$	1.10626e8	1.02299	0.511495	+	0.859286i	$0.329092\pi$
$62$	0	0
$63$	−2.67465e8	−2.13912
$64$	0	0
$65$	1.39662e7	0.0970440
$66$	0	0
$67$	1.26183e8	0.765007	0.382503	−	0.923954i	$-0.375062\pi$
$67$	1.26183e8	0.765007	0.382503	+	0.923954i	$0.375062\pi$
$68$	0	0
$69$	−6.81323e7	−0.361853
$70$	0	0
$71$	1.44850e8	0.676480	0.338240	−	0.941060i	$-0.390168\pi$
$71$	1.44850e8	0.676480	0.338240	+	0.941060i	$0.390168\pi$
$72$	0	0
$73$	−3.98283e7	−0.164149	−0.0820747	−	0.996626i	$-0.526155\pi$
$73$	−3.98283e7	−0.164149	−0.0820747	+	0.996626i	$0.526155\pi$
$74$	0	0
$75$	−3.59434e8	−1.31173
$76$	0	0
$77$	2.59632e8	0.841684
$78$	0	0
$79$	−2.09504e8	−0.605159	−0.302579	−	0.953124i	$-0.597848\pi$
$79$	−2.09504e8	−0.605159	−0.302579	+	0.953124i	$0.597848\pi$
$80$	0	0
$81$	−2.75441e8	−0.710963
$82$	0	0
$83$	−1.27715e8	−0.295387	−0.147694	−	0.989033i	$-0.547185\pi$
$83$	−1.27715e8	−0.295387	−0.147694	+	0.989033i	$0.547185\pi$
$84$	0	0
$85$	7.85822e7	0.163282
$86$	0	0
$87$	9.54482e8	1.78620
$88$	0	0
$89$	−1.03596e9	−1.75020	−0.875098	−	0.483947i	$-0.839203\pi$
$89$	−1.03596e9	−1.75020	−0.875098	+	0.483947i	$0.839203\pi$
$90$	0	0
$91$	−3.14461e8	−0.480707
$92$	0	0
$93$	2.03017e9	2.81422
$94$	0	0
$95$	1.94536e8	0.245044
$96$	0	0
$97$	1.07626e9	1.23437	0.617185	−	0.786818i	$-0.288273\pi$
$97$	1.07626e9	1.23437	0.617185	+	0.786818i	$0.288273\pi$
$98$	0	0
$99$	−5.72846e8	−0.599349
$100$	0	0
$101$	4.50063e8	0.430355	0.215178	−	0.976575i	$-0.430967\pi$
$101$	4.50063e8	0.430355	0.215178	+	0.976575i	$0.430967\pi$
$102$	0	0
$103$	−1.19015e9	−1.04192	−0.520960	−	0.853581i	$-0.674426\pi$
$103$	−1.19015e9	−1.04192	−0.520960	+	0.853581i	$0.674426\pi$
$104$	0	0
$105$	−1.12903e9	−0.906469
$106$	0	0
$107$	−8.89269e8	−0.655852	−0.327926	−	0.944703i	$-0.606350\pi$
$107$	−8.89269e8	−0.655852	−0.327926	+	0.944703i	$0.606350\pi$
$108$	0	0
$109$	2.28855e9	1.55289	0.776446	−	0.630184i	$-0.217020\pi$
$109$	2.28855e9	1.55289	0.776446	+	0.630184i	$0.217020\pi$
$110$	0	0
$111$	1.19692e9	0.748362
$112$	0	0
$113$	1.84249e9	1.06305	0.531524	−	0.847043i	$-0.321620\pi$
$113$	1.84249e9	1.06305	0.531524	+	0.847043i	$0.321620\pi$
$114$	0	0
$115$	−1.58874e8	−0.0847056
$116$	0	0
$117$	6.93821e8	0.342303
$118$	0	0
$119$	−1.76934e9	−0.808818
$120$	0	0
$121$	−1.80188e9	−0.764173
$122$	0	0
$123$	2.30875e9	0.909504
$124$	0	0
$125$	−1.79321e9	−0.656957
$126$	0	0
$127$	−5.07757e9	−1.73196	−0.865982	−	0.500075i	$-0.833306\pi$
$127$	−5.07757e9	−1.73196	−0.865982	+	0.500075i	$0.833306\pi$
$128$	0	0
$129$	−4.29965e9	−1.36703
$130$	0	0
$131$	2.40078e9	0.712248	0.356124	−	0.934439i	$-0.384098\pi$
$131$	2.40078e9	0.712248	0.356124	+	0.934439i	$0.384098\pi$
$132$	0	0
$133$	−4.38015e9	−1.21383
$134$	0	0
$135$	4.72687e8	0.122482
$136$	0	0
$137$	−2.13089e9	−0.516795	−0.258397	−	0.966039i	$-0.583194\pi$
$137$	−2.13089e9	−0.516795	−0.258397	+	0.966039i	$0.583194\pi$
$138$	0	0
$139$	8.96667e8	0.203735	0.101867	−	0.994798i	$-0.467518\pi$
$139$	8.96667e8	0.203735	0.101867	+	0.994798i	$0.467518\pi$
$140$	0	0
$141$	9.69215e9	2.06507
$142$	0	0
$143$	−6.73500e8	−0.134687
$144$	0	0
$145$	2.22570e9	0.418130
$146$	0	0
$147$	1.69587e10	2.99547
$148$	0	0
$149$	−5.19244e8	−0.0863044	−0.0431522	−	0.999069i	$-0.513740\pi$
$149$	−5.19244e8	−0.0863044	−0.0431522	+	0.999069i	$0.513740\pi$
$150$	0	0
$151$	5.64069e9	0.882950	0.441475	−	0.897274i	$-0.354455\pi$
$151$	5.64069e9	0.882950	0.441475	+	0.897274i	$0.354455\pi$
$152$	0	0
$153$	3.90385e9	0.575945
$154$	0	0
$155$	4.73404e9	0.658778
$156$	0	0
$157$	7.87589e9	1.03455	0.517275	−	0.855819i	$-0.326947\pi$
$157$	7.87589e9	1.03455	0.517275	+	0.855819i	$0.326947\pi$
$158$	0	0
$159$	1.99485e10	2.47527
$160$	0	0
$161$	3.57718e9	0.419589
$162$	0	0
$163$	1.14373e10	1.26905	0.634524	−	0.772903i	$-0.281196\pi$
$163$	1.14373e10	1.26905	0.634524	+	0.772903i	$0.281196\pi$
$164$	0	0
$165$	−2.41811e9	−0.253979
$166$	0	0
$167$	2.85367e8	0.0283909	0.0141955	−	0.999899i	$-0.495481\pi$
$167$	2.85367e8	0.0283909	0.0141955	+	0.999899i	$0.495481\pi$
$168$	0	0
$169$	8.15731e8	0.0769231
$170$	0	0
$171$	9.66428e9	0.864344
$172$	0	0
$173$	−5.16392e9	−0.438300	−0.219150	−	0.975691i	$-0.570328\pi$
$173$	−5.16392e9	−0.438300	−0.219150	+	0.975691i	$0.570328\pi$
$174$	0	0
$175$	1.88715e10	1.52102
$176$	0	0
$177$	−4.22276e9	−0.323382
$178$	0	0
$179$	−7.64026e9	−0.556249	−0.278125	−	0.960545i	$-0.589713\pi$
$179$	−7.64026e9	−0.556249	−0.278125	+	0.960545i	$0.589713\pi$
$180$	0	0
$181$	2.03632e10	1.41024	0.705118	−	0.709090i	$-0.250894\pi$
$181$	2.03632e10	1.41024	0.705118	+	0.709090i	$0.250894\pi$
$182$	0	0
$183$	2.31986e10	1.52908
$184$	0	0
$185$	2.79103e9	0.175183
$186$	0	0
$187$	−3.78951e9	−0.226619
$188$	0	0
$189$	−1.06429e10	−0.606714
$190$	0	0
$191$	−2.20144e10	−1.19690	−0.598448	−	0.801161i	$-0.704216\pi$
$191$	−2.20144e10	−1.19690	−0.598448	+	0.801161i	$0.704216\pi$
$192$	0	0
$193$	1.87942e10	0.975024	0.487512	−	0.873116i	$-0.337904\pi$
$193$	1.87942e10	0.975024	0.487512	+	0.873116i	$0.337904\pi$
$194$	0	0
$195$	2.92877e9	0.145054
$196$	0	0
$197$	1.52558e10	0.721668	0.360834	−	0.932630i	$-0.382492\pi$
$197$	1.52558e10	0.721668	0.360834	+	0.932630i	$0.382492\pi$
$198$	0	0
$199$	−2.41160e10	−1.09010	−0.545049	−	0.838404i	$-0.683489\pi$
$199$	−2.41160e10	−1.09010	−0.545049	+	0.838404i	$0.683489\pi$
$200$	0	0
$201$	2.64611e10	1.14347
$202$	0	0
$203$	−5.01135e10	−2.07120
$204$	0	0
$205$	5.38366e9	0.212905
$206$	0	0
$207$	−7.89261e9	−0.298782
$208$	0	0
$209$	−9.38123e9	−0.340096
$210$	0	0
$211$	9.12569e9	0.316953	0.158476	−	0.987363i	$-0.449342\pi$
$211$	9.12569e9	0.316953	0.158476	+	0.987363i	$0.449342\pi$
$212$	0	0
$213$	3.03755e10	1.01115
$214$	0	0
$215$	−1.00261e10	−0.320007
$216$	0	0
$217$	−1.06591e11	−3.26325
$218$	0	0
$219$	−8.35214e9	−0.245357
$220$	0	0
$221$	4.58978e9	0.129428
$222$	0	0
$223$	4.20762e10	1.13937	0.569684	−	0.821864i	$-0.307065\pi$
$223$	4.20762e10	1.13937	0.569684	+	0.821864i	$0.307065\pi$
$224$	0	0
$225$	−4.16377e10	−1.08309
$226$	0	0
$227$	2.18393e10	0.545912	0.272956	−	0.962027i	$-0.411999\pi$
$227$	2.18393e10	0.545912	0.272956	+	0.962027i	$0.411999\pi$
$228$	0	0
$229$	4.18787e10	1.00631	0.503157	−	0.864195i	$-0.332172\pi$
$229$	4.18787e10	1.00631	0.503157	+	0.864195i	$0.332172\pi$
$230$	0	0
$231$	5.44457e10	1.25808
$232$	0	0
$233$	1.19849e10	0.266399	0.133199	−	0.991089i	$-0.457475\pi$
$233$	1.19849e10	0.266399	0.133199	+	0.991089i	$0.457475\pi$
$234$	0	0
$235$	2.26006e10	0.483409
$236$	0	0
$237$	−4.39336e10	−0.904544
$238$	0	0
$239$	−6.91323e10	−1.37054	−0.685268	−	0.728291i	$-0.740315\pi$
$239$	−6.91323e10	−1.37054	−0.685268	+	0.728291i	$0.740315\pi$
$240$	0	0
$241$	7.97901e10	1.52361	0.761803	−	0.647809i	$-0.224315\pi$
$241$	7.97901e10	1.52361	0.761803	+	0.647809i	$0.224315\pi$
$242$	0	0
$243$	−7.67876e10	−1.41274
$244$	0	0
$245$	3.95450e10	0.701204
$246$	0	0
$247$	1.13624e10	0.194237
$248$	0	0
$249$	−2.67824e10	−0.441522
$250$	0	0
$251$	2.43450e10	0.387150	0.193575	−	0.981086i	$-0.437992\pi$
$251$	2.43450e10	0.387150	0.193575	+	0.981086i	$0.437992\pi$
$252$	0	0
$253$	7.66145e9	0.117562
$254$	0	0
$255$	1.64790e10	0.244062
$256$	0	0
$257$	−7.83619e10	−1.12048	−0.560242	−	0.828329i	$-0.689292\pi$
$257$	−7.83619e10	−1.12048	−0.560242	+	0.828329i	$0.689292\pi$
$258$	0	0
$259$	−6.28424e10	−0.867768
$260$	0	0
$261$	1.10570e11	1.47487
$262$	0	0
$263$	−5.44846e10	−0.702219	−0.351110	−	0.936334i	$-0.614196\pi$
$263$	−5.44846e10	−0.702219	−0.351110	+	0.936334i	$0.614196\pi$
$264$	0	0
$265$	4.65168e10	0.579433
$266$	0	0
$267$	−2.17244e11	−2.61605
$268$	0	0
$269$	−1.66373e10	−0.193730	−0.0968651	−	0.995298i	$-0.530882\pi$
$269$	−1.66373e10	−0.193730	−0.0968651	+	0.995298i	$0.530882\pi$
$270$	0	0
$271$	−1.35918e11	−1.53079	−0.765394	−	0.643562i	$-0.777456\pi$
$271$	−1.35918e11	−1.53079	−0.765394	+	0.643562i	$0.777456\pi$
$272$	0	0
$273$	−6.59436e10	−0.718523
$274$	0	0
$275$	4.04182e10	0.426167
$276$	0	0
$277$	1.04715e11	1.06869	0.534345	−	0.845266i	$-0.320558\pi$
$277$	1.04715e11	1.06869	0.534345	+	0.845266i	$0.320558\pi$
$278$	0	0
$279$	2.35180e11	2.32371
$280$	0	0
$281$	−1.75058e11	−1.67495	−0.837477	−	0.546472i	$-0.815971\pi$
$281$	−1.75058e11	−1.67495	−0.837477	+	0.546472i	$0.815971\pi$
$282$	0	0
$283$	−7.71468e10	−0.714955	−0.357478	−	0.933922i	$-0.616363\pi$
$283$	−7.71468e10	−0.714955	−0.357478	+	0.933922i	$0.616363\pi$
$284$	0	0
$285$	4.07950e10	0.366273
$286$	0	0
$287$	−1.21217e11	−1.05462
$288$	0	0
$289$	−9.27630e10	−0.782230
$290$	0	0
$291$	2.25696e11	1.84504
$292$	0	0
$293$	1.92868e11	1.52882	0.764410	−	0.644730i	$-0.223030\pi$
$293$	1.92868e11	1.52882	0.764410	+	0.644730i	$0.223030\pi$
$294$	0	0
$295$	−9.84681e9	−0.0757001
$296$	0	0
$297$	−2.27946e10	−0.169992
$298$	0	0
$299$	−9.27941e9	−0.0671429
$300$	0	0
$301$	2.25746e11	1.58515
$302$	0	0
$303$	9.43798e10	0.643261
$304$	0	0
$305$	5.40955e10	0.357941
$306$	0	0
$307$	3.50351e10	0.225103	0.112551	−	0.993646i	$-0.464098\pi$
$307$	3.50351e10	0.225103	0.112551	+	0.993646i	$0.464098\pi$
$308$	0	0
$309$	−2.49579e11	−1.55738
$310$	0	0
$311$	−3.10470e11	−1.88190	−0.940952	−	0.338540i	$-0.890067\pi$
$311$	−3.10470e11	−1.88190	−0.940952	+	0.338540i	$0.890067\pi$
$312$	0	0
$313$	1.83696e11	1.08181	0.540905	−	0.841084i	$-0.318082\pi$
$313$	1.83696e11	1.08181	0.540905	+	0.841084i	$0.318082\pi$
$314$	0	0
$315$	−1.30789e11	−0.748471
$316$	0	0
$317$	−2.20565e11	−1.22679	−0.613394	−	0.789777i	$-0.710196\pi$
$317$	−2.20565e11	−1.22679	−0.613394	+	0.789777i	$0.710196\pi$
$318$	0	0
$319$	−1.07331e11	−0.580320
$320$	0	0
$321$	−1.86483e11	−0.980316
$322$	0	0
$323$	6.39314e10	0.326816
$324$	0	0
$325$	−4.89538e10	−0.243395
$326$	0	0
$327$	4.79917e11	2.32114
$328$	0	0
$329$	−5.08871e11	−2.39456
$330$	0	0
$331$	4.51234e10	0.206621	0.103311	−	0.994649i	$-0.467056\pi$
$331$	4.51234e10	0.206621	0.103311	+	0.994649i	$0.467056\pi$
$332$	0	0
$333$	1.38654e11	0.617923
$334$	0	0
$335$	6.17031e10	0.267674
$336$	0	0
$337$	1.61655e11	0.682738	0.341369	−	0.939929i	$-0.389109\pi$
$337$	1.61655e11	0.682738	0.341369	+	0.939929i	$0.389109\pi$
$338$	0	0
$339$	3.86377e11	1.58896
$340$	0	0
$341$	−2.28292e11	−0.914315
$342$	0	0
$343$	−4.46090e11	−1.74020
$344$	0	0
$345$	−3.33164e10	−0.126611
$346$	0	0
$347$	3.66990e11	1.35885	0.679425	−	0.733745i	$-0.262229\pi$
$347$	3.66990e11	1.35885	0.679425	+	0.733745i	$0.262229\pi$
$348$	0	0
$349$	−5.29964e11	−1.91219	−0.956096	−	0.293053i	$-0.905329\pi$
$349$	−5.29964e11	−1.91219	−0.956096	+	0.293053i	$0.905329\pi$
$350$	0	0
$351$	2.76084e10	0.0970867
$352$	0	0
$353$	1.77519e11	0.608497	0.304249	−	0.952593i	$-0.401595\pi$
$353$	1.77519e11	0.608497	0.304249	+	0.952593i	$0.401595\pi$
$354$	0	0
$355$	7.08309e10	0.236698
$356$	0	0
$357$	−3.71038e11	−1.20896
$358$	0	0
$359$	5.57397e11	1.77109	0.885544	−	0.464556i	$-0.153786\pi$
$359$	5.57397e11	1.77109	0.885544	+	0.464556i	$0.153786\pi$
$360$	0	0
$361$	−1.64421e11	−0.509535
$362$	0	0
$363$	−3.77860e11	−1.14223
$364$	0	0
$365$	−1.94759e10	−0.0574354
$366$	0	0
$367$	1.20340e11	0.346269	0.173134	−	0.984898i	$-0.444611\pi$
$367$	1.20340e11	0.346269	0.173134	+	0.984898i	$0.444611\pi$
$368$	0	0
$369$	2.67452e11	0.750978
$370$	0	0
$371$	−1.04736e12	−2.87022
$372$	0	0
$373$	−7.03469e11	−1.88172	−0.940861	−	0.338793i	$-0.889981\pi$
$373$	−7.03469e11	−1.88172	−0.940861	+	0.338793i	$0.889981\pi$
$374$	0	0
$375$	−3.76043e11	−0.981968
$376$	0	0
$377$	1.29998e11	0.331436
$378$	0	0
$379$	−5.50542e11	−1.37061	−0.685306	−	0.728256i	$-0.740331\pi$
$379$	−5.50542e11	−1.37061	−0.685306	+	0.728256i	$0.740331\pi$
$380$	0	0
$381$	−1.06478e12	−2.58880
$382$	0	0
$383$	8.34780e11	1.98234	0.991168	−	0.132612i	$-0.0423363\pi$
$383$	8.34780e11	1.98234	0.991168	+	0.132612i	$0.0423363\pi$
$384$	0	0
$385$	1.26959e11	0.294503
$386$	0	0
$387$	−4.98082e11	−1.12876
$388$	0	0
$389$	7.60130e11	1.68312	0.841559	−	0.540164i	$-0.181638\pi$
$389$	7.60130e11	1.68312	0.841559	+	0.540164i	$0.181638\pi$
$390$	0	0
$391$	−5.22114e10	−0.112972
$392$	0	0
$393$	5.03451e11	1.06461
$394$	0	0
$395$	−1.02446e11	−0.211743
$396$	0	0
$397$	−1.72936e11	−0.349405	−0.174702	−	0.984621i	$-0.555896\pi$
$397$	−1.72936e11	−0.349405	−0.174702	+	0.984621i	$0.555896\pi$
$398$	0	0
$399$	−9.18533e11	−1.81433
$400$	0	0
$401$	5.85244e11	1.13028	0.565141	−	0.824994i	$-0.308822\pi$
$401$	5.85244e11	1.13028	0.565141	+	0.824994i	$0.308822\pi$
$402$	0	0
$403$	2.76503e11	0.522188
$404$	0	0
$405$	−1.34690e11	−0.248764
$406$	0	0
$407$	−1.34593e11	−0.243136
$408$	0	0
$409$	−1.90539e11	−0.336689	−0.168344	−	0.985728i	$-0.553842\pi$
$409$	−1.90539e11	−0.336689	−0.168344	+	0.985728i	$0.553842\pi$
$410$	0	0
$411$	−4.46855e11	−0.772464
$412$	0	0
$413$	2.21709e11	0.374980
$414$	0	0
$415$	−6.24523e10	−0.103355
$416$	0	0
$417$	1.88034e11	0.304526
$418$	0	0
$419$	9.62485e11	1.52557	0.762783	−	0.646655i	$-0.223833\pi$
$419$	9.62485e11	1.52557	0.762783	+	0.646655i	$0.223833\pi$
$420$	0	0
$421$	3.13093e11	0.485740	0.242870	−	0.970059i	$-0.421911\pi$
$421$	3.13093e11	0.485740	0.242870	+	0.970059i	$0.421911\pi$
$422$	0	0
$423$	1.12276e12	1.70513
$424$	0	0
$425$	−2.75443e11	−0.409526
$426$	0	0
$427$	−1.21800e12	−1.77306
$428$	0	0
$429$	−1.41235e11	−0.201319
$430$	0	0
$431$	−8.35637e11	−1.16646	−0.583230	−	0.812307i	$-0.698211\pi$
$431$	−8.35637e11	−1.16646	−0.583230	+	0.812307i	$0.698211\pi$
$432$	0	0
$433$	3.32262e11	0.454239	0.227120	−	0.973867i	$-0.427069\pi$
$433$	3.32262e11	0.454239	0.227120	+	0.973867i	$0.427069\pi$
$434$	0	0
$435$	4.66738e11	0.624988
$436$	0	0
$437$	−1.29253e11	−0.169541
$438$	0	0
$439$	8.43501e11	1.08391	0.541957	−	0.840406i	$-0.317683\pi$
$439$	8.43501e11	1.08391	0.541957	+	0.840406i	$0.317683\pi$
$440$	0	0
$441$	1.96454e12	2.47336
$442$	0	0
$443$	7.60339e10	0.0937973	0.0468986	−	0.998900i	$-0.485066\pi$
$443$	7.60339e10	0.0937973	0.0468986	+	0.998900i	$0.485066\pi$
$444$	0	0
$445$	−5.06579e11	−0.612388
$446$	0	0
$447$	−1.08887e11	−0.129001
$448$	0	0
$449$	−5.94316e11	−0.690095	−0.345047	−	0.938585i	$-0.612137\pi$
$449$	−5.94316e11	−0.690095	−0.345047	+	0.938585i	$0.612137\pi$
$450$	0	0
$451$	−2.59619e11	−0.295489
$452$	0	0
$453$	1.18287e12	1.31976
$454$	0	0
$455$	−1.53770e11	−0.168198
$456$	0	0
$457$	6.46986e11	0.693861	0.346930	−	0.937891i	$-0.387224\pi$
$457$	6.46986e11	0.693861	0.346930	+	0.937891i	$0.387224\pi$
$458$	0	0
$459$	1.55341e11	0.163354
$460$	0	0
$461$	2.49748e10	0.0257542	0.0128771	−	0.999917i	$-0.495901\pi$
$461$	2.49748e10	0.0257542	0.0128771	+	0.999917i	$0.495901\pi$
$462$	0	0
$463$	1.86304e12	1.88412	0.942058	−	0.335451i	$-0.108889\pi$
$463$	1.86304e12	1.88412	0.942058	+	0.335451i	$0.108889\pi$
$464$	0	0
$465$	9.92745e11	0.984689
$466$	0	0
$467$	1.93423e12	1.88184	0.940918	−	0.338635i	$-0.109965\pi$
$467$	1.93423e12	1.88184	0.940918	+	0.338635i	$0.109965\pi$
$468$	0	0
$469$	−1.38930e12	−1.32592
$470$	0	0
$471$	1.65160e12	1.54636
$472$	0	0
$473$	4.83494e11	0.444136
$474$	0	0
$475$	−6.81880e11	−0.614592
$476$	0	0
$477$	2.31088e12	2.04383
$478$	0	0
$479$	−1.24141e12	−1.07747	−0.538735	−	0.842475i	$-0.681097\pi$
$479$	−1.24141e12	−1.07747	−0.538735	+	0.842475i	$0.681097\pi$
$480$	0	0
$481$	1.63017e11	0.138861
$482$	0	0
$483$	7.50146e11	0.627168
$484$	0	0
$485$	5.26288e11	0.431902
$486$	0	0
$487$	−9.20890e11	−0.741869	−0.370935	−	0.928659i	$-0.620963\pi$
$487$	−9.20890e11	−0.741869	−0.370935	+	0.928659i	$0.620963\pi$
$488$	0	0
$489$	2.39844e12	1.89687
$490$	0	0
$491$	−1.07106e12	−0.831661	−0.415831	−	0.909442i	$-0.636509\pi$
$491$	−1.07106e12	−0.831661	−0.415831	+	0.909442i	$0.636509\pi$
$492$	0	0
$493$	7.31443e11	0.557660
$494$	0	0
$495$	−2.80120e11	−0.209711
$496$	0	0
$497$	−1.59482e12	−1.17248
$498$	0	0
$499$	−1.38423e11	−0.0999438	−0.0499719	−	0.998751i	$-0.515913\pi$
$499$	−1.38423e11	−0.0999438	−0.0499719	+	0.998751i	$0.515913\pi$
$500$	0	0
$501$	5.98425e10	0.0424365
$502$	0	0
$503$	2.40845e12	1.67757	0.838785	−	0.544462i	$-0.183266\pi$
$503$	2.40845e12	1.67757	0.838785	+	0.544462i	$0.183266\pi$
$504$	0	0
$505$	2.20079e11	0.150580
$506$	0	0
$507$	1.71062e11	0.114979
$508$	0	0
$509$	−1.30609e12	−0.862468	−0.431234	−	0.902240i	$-0.641922\pi$
$509$	−1.30609e12	−0.862468	−0.431234	+	0.902240i	$0.641922\pi$
$510$	0	0
$511$	4.38516e11	0.284506
$512$	0	0
$513$	3.84560e11	0.245152
$514$	0	0
$515$	−5.81979e11	−0.364565
$516$	0	0
$517$	−1.08988e12	−0.670921
$518$	0	0
$519$	−1.08289e12	−0.655137
$520$	0	0
$521$	−1.17098e12	−0.696274	−0.348137	−	0.937444i	$-0.613186\pi$
$521$	−1.17098e12	−0.696274	−0.348137	+	0.937444i	$0.613186\pi$
$522$	0	0
$523$	−7.76176e11	−0.453631	−0.226815	−	0.973938i	$-0.572831\pi$
$523$	−7.76176e11	−0.453631	−0.226815	+	0.973938i	$0.572831\pi$
$524$	0	0
$525$	3.95742e12	2.27350
$526$	0	0
$527$	1.55577e12	0.878612
$528$	0	0
$529$	−1.69559e12	−0.941394
$530$	0	0
$531$	−4.89175e11	−0.267017
$532$	0	0
$533$	3.14445e11	0.168761
$534$	0	0
$535$	−4.34849e11	−0.229481
$536$	0	0
$537$	−1.60219e12	−0.831437
$538$	0	0
$539$	−1.90700e12	−0.973198
$540$	0	0
$541$	2.59227e11	0.130105	0.0650523	−	0.997882i	$-0.479279\pi$
$541$	2.59227e11	0.130105	0.0650523	+	0.997882i	$0.479279\pi$
$542$	0	0
$543$	4.27023e12	2.10791
$544$	0	0
$545$	1.11909e12	0.543353
$546$	0	0
$547$	−1.74086e12	−0.831421	−0.415710	−	0.909497i	$-0.636467\pi$
$547$	−1.74086e12	−0.831421	−0.415710	+	0.909497i	$0.636467\pi$
$548$	0	0
$549$	2.68738e12	1.26257
$550$	0	0
$551$	1.81074e12	0.836903
$552$	0	0
$553$	2.30667e12	1.04887
$554$	0	0
$555$	5.85289e11	0.261850
$556$	0	0
$557$	−4.29154e12	−1.88914	−0.944571	−	0.328309i	$-0.893521\pi$
$557$	−4.29154e12	−1.88914	−0.944571	+	0.328309i	$0.893521\pi$
$558$	0	0
$559$	−5.85599e11	−0.253657
$560$	0	0
$561$	−7.94674e11	−0.338732
$562$	0	0
$563$	−3.14494e12	−1.31924	−0.659621	−	0.751598i	$-0.729283\pi$
$563$	−3.14494e12	−1.31924	−0.659621	+	0.751598i	$0.729283\pi$
$564$	0	0
$565$	9.00972e11	0.371958
$566$	0	0
$567$	3.03265e12	1.23225
$568$	0	0
$569$	−2.38655e12	−0.954475	−0.477238	−	0.878774i	$-0.658362\pi$
$569$	−2.38655e12	−0.954475	−0.477238	+	0.878774i	$0.658362\pi$
$570$	0	0
$571$	−2.29305e12	−0.902718	−0.451359	−	0.892343i	$-0.649061\pi$
$571$	−2.29305e12	−0.902718	−0.451359	+	0.892343i	$0.649061\pi$
$572$	0	0
$573$	−4.61650e12	−1.78903
$574$	0	0
$575$	5.56878e11	0.212449
$576$	0	0
$577$	−2.28430e12	−0.857950	−0.428975	−	0.903316i	$-0.641125\pi$
$577$	−2.28430e12	−0.857950	−0.428975	+	0.903316i	$0.641125\pi$
$578$	0	0
$579$	3.94121e12	1.45739
$580$	0	0
$581$	1.40617e12	0.511969
$582$	0	0
$583$	−2.24320e12	−0.804192
$584$	0	0
$585$	3.39276e11	0.119771
$586$	0	0
$587$	2.41317e12	0.838912	0.419456	−	0.907776i	$-0.362221\pi$
$587$	2.41317e12	0.838912	0.419456	+	0.907776i	$0.362221\pi$
$588$	0	0
$589$	3.85143e12	1.31857
$590$	0	0
$591$	3.19920e12	1.07869
$592$	0	0
$593$	−3.27743e12	−1.08840	−0.544199	−	0.838956i	$-0.683166\pi$
$593$	−3.27743e12	−1.08840	−0.544199	+	0.838956i	$0.683166\pi$
$594$	0	0
$595$	−8.65202e11	−0.283003
$596$	0	0
$597$	−5.05720e12	−1.62939
$598$	0	0
$599$	−4.38370e12	−1.39130	−0.695650	−	0.718381i	$-0.744883\pi$
$599$	−4.38370e12	−1.39130	−0.695650	+	0.718381i	$0.744883\pi$
$600$	0	0
$601$	2.93453e12	0.917494	0.458747	−	0.888567i	$-0.348298\pi$
$601$	2.93453e12	0.917494	0.458747	+	0.888567i	$0.348298\pi$
$602$	0	0
$603$	3.06532e12	0.944165
$604$	0	0
$605$	−8.81112e11	−0.267382
$606$	0	0
$607$	−3.73411e12	−1.11645	−0.558224	−	0.829690i	$-0.688517\pi$
$607$	−3.73411e12	−1.11645	−0.558224	+	0.829690i	$0.688517\pi$
$608$	0	0
$609$	−1.05090e13	−3.09587
$610$	0	0
$611$	1.32004e12	0.383180
$612$	0	0
$613$	4.86942e11	0.139285	0.0696427	−	0.997572i	$-0.477814\pi$
$613$	4.86942e11	0.139285	0.0696427	+	0.997572i	$0.477814\pi$
$614$	0	0
$615$	1.12897e12	0.318233
$616$	0	0
$617$	−4.58538e12	−1.27377	−0.636887	−	0.770957i	$-0.719778\pi$
$617$	−4.58538e12	−1.27377	−0.636887	+	0.770957i	$0.719778\pi$
$618$	0	0
$619$	2.33142e12	0.638281	0.319141	−	0.947707i	$-0.396606\pi$
$619$	2.33142e12	0.638281	0.319141	+	0.947707i	$0.396606\pi$
$620$	0	0
$621$	−3.14062e11	−0.0847429
$622$	0	0
$623$	1.14060e13	3.03346
$624$	0	0
$625$	2.47080e12	0.647705
$626$	0	0
$627$	−1.96728e12	−0.508348
$628$	0	0
$629$	9.17229e11	0.233642
$630$	0	0
$631$	1.23807e12	0.310895	0.155448	−	0.987844i	$-0.450318\pi$
$631$	1.23807e12	0.310895	0.155448	+	0.987844i	$0.450318\pi$
$632$	0	0
$633$	1.91369e12	0.473756
$634$	0	0
$635$	−2.48291e12	−0.606009
$636$	0	0
$637$	2.30972e12	0.555818
$638$	0	0
$639$	3.51877e12	0.834906
$640$	0	0
$641$	8.11886e11	0.189948	0.0949738	−	0.995480i	$-0.469723\pi$
$641$	8.11886e11	0.189948	0.0949738	+	0.995480i	$0.469723\pi$
$642$	0	0
$643$	−3.19368e12	−0.736788	−0.368394	−	0.929670i	$-0.620092\pi$
$643$	−3.19368e12	−0.736788	−0.368394	+	0.929670i	$0.620092\pi$
$644$	0	0
$645$	−2.10251e12	−0.478321
$646$	0	0
$647$	9.88017e11	0.221664	0.110832	−	0.993839i	$-0.464648\pi$
$647$	9.88017e11	0.221664	0.110832	+	0.993839i	$0.464648\pi$
$648$	0	0
$649$	4.74848e11	0.105064
$650$	0	0
$651$	−2.23525e13	−4.87765
$652$	0	0
$653$	2.31322e12	0.497860	0.248930	−	0.968521i	$-0.419921\pi$
$653$	2.31322e12	0.497860	0.248930	+	0.968521i	$0.419921\pi$
$654$	0	0
$655$	1.17397e12	0.249213
$656$	0	0
$657$	−9.67533e11	−0.202592
$658$	0	0
$659$	−8.88926e12	−1.83604	−0.918018	−	0.396539i	$-0.870211\pi$
$659$	−8.88926e12	−1.83604	−0.918018	+	0.396539i	$0.870211\pi$
$660$	0	0
$661$	−3.75968e12	−0.766028	−0.383014	−	0.923742i	$-0.625114\pi$
$661$	−3.75968e12	−0.766028	−0.383014	+	0.923742i	$0.625114\pi$
$662$	0	0
$663$	9.62494e11	0.193458
$664$	0	0
$665$	−2.14188e12	−0.424714
$666$	0	0
$667$	−1.47880e12	−0.289296
$668$	0	0
$669$	8.82352e12	1.70304
$670$	0	0
$671$	−2.60867e12	−0.496785
$672$	0	0
$673$	2.37969e11	0.0447150	0.0223575	−	0.999750i	$-0.492883\pi$
$673$	2.37969e11	0.0447150	0.0223575	+	0.999750i	$0.492883\pi$
$674$	0	0
$675$	−1.65684e12	−0.307195
$676$	0	0
$677$	−9.42990e11	−0.172527	−0.0862637	−	0.996272i	$-0.527493\pi$
$677$	−9.42990e11	−0.172527	−0.0862637	+	0.996272i	$0.527493\pi$
$678$	0	0
$679$	−1.18498e13	−2.13943
$680$	0	0
$681$	4.57978e12	0.815986
$682$	0	0
$683$	2.87035e12	0.504709	0.252355	−	0.967635i	$-0.418795\pi$
$683$	2.87035e12	0.504709	0.252355	+	0.967635i	$0.418795\pi$
$684$	0	0
$685$	−1.04200e12	−0.180825
$686$	0	0
$687$	8.78211e12	1.50416
$688$	0	0
$689$	2.71692e12	0.459294
$690$	0	0
$691$	−5.99596e12	−1.00048	−0.500239	−	0.865887i	$-0.666754\pi$
$691$	−5.99596e12	−1.00048	−0.500239	+	0.865887i	$0.666754\pi$
$692$	0	0
$693$	6.30712e12	1.03880
$694$	0	0
$695$	4.38467e11	0.0712862
$696$	0	0
$697$	1.76926e12	0.283951
$698$	0	0
$699$	2.51327e12	0.398192
$700$	0	0
$701$	8.08801e12	1.26506	0.632530	−	0.774536i	$-0.282017\pi$
$701$	8.08801e12	1.26506	0.632530	+	0.774536i	$0.282017\pi$
$702$	0	0
$703$	2.27067e12	0.350635
$704$	0	0
$705$	4.73943e12	0.722561
$706$	0	0
$707$	−4.95526e12	−0.745898
$708$	0	0
$709$	5.55768e12	0.826011	0.413005	−	0.910729i	$-0.364479\pi$
$709$	5.55768e12	0.826011	0.413005	+	0.910729i	$0.364479\pi$
$710$	0	0
$711$	−5.08938e12	−0.746882
$712$	0	0
$713$	−3.14538e12	−0.455796
$714$	0	0
$715$	−3.29339e11	−0.0471266
$716$	0	0
$717$	−1.44973e13	−2.04857
$718$	0	0
$719$	5.71213e12	0.797109	0.398555	−	0.917145i	$-0.369512\pi$
$719$	5.71213e12	0.797109	0.398555	+	0.917145i	$0.369512\pi$
$720$	0	0
$721$	1.31037e13	1.80587
$722$	0	0
$723$	1.67323e13	2.27737
$724$	0	0
$725$	−7.80144e12	−1.04871
$726$	0	0
$727$	2.04269e12	0.271205	0.135603	−	0.990763i	$-0.456703\pi$
$727$	2.04269e12	0.271205	0.135603	+	0.990763i	$0.456703\pi$
$728$	0	0
$729$	−1.06811e13	−1.40069
$730$	0	0
$731$	−3.29493e12	−0.426793
$732$	0	0
$733$	−1.18481e12	−0.151594	−0.0757971	−	0.997123i	$-0.524150\pi$
$733$	−1.18481e12	−0.151594	−0.0757971	+	0.997123i	$0.524150\pi$
$734$	0	0
$735$	8.29274e12	1.04811
$736$	0	0
$737$	−2.97554e12	−0.371503
$738$	0	0
$739$	8.43689e12	1.04060	0.520298	−	0.853985i	$-0.325821\pi$
$739$	8.43689e12	1.04060	0.520298	+	0.853985i	$0.325821\pi$
$740$	0	0
$741$	2.38273e12	0.290331
$742$	0	0
$743$	−7.29494e12	−0.878157	−0.439079	−	0.898449i	$-0.644695\pi$
$743$	−7.29494e12	−0.878157	−0.439079	+	0.898449i	$0.644695\pi$
$744$	0	0
$745$	−2.53908e11	−0.0301977
$746$	0	0
$747$	−3.10254e12	−0.364565
$748$	0	0
$749$	9.79098e12	1.13673
$750$	0	0
$751$	−2.29023e12	−0.262724	−0.131362	−	0.991334i	$-0.541935\pi$
$751$	−2.29023e12	−0.262724	−0.131362	+	0.991334i	$0.541935\pi$
$752$	0	0
$753$	5.10524e12	0.578681
$754$	0	0
$755$	2.75828e12	0.308942
$756$	0	0
$757$	−2.83841e12	−0.314155	−0.157078	−	0.987586i	$-0.550207\pi$
$757$	−2.83841e12	−0.314155	−0.157078	+	0.987586i	$0.550207\pi$
$758$	0	0
$759$	1.60663e12	0.175723
$760$	0	0
$761$	−1.70378e13	−1.84155	−0.920774	−	0.390096i	$-0.872442\pi$
$761$	−1.70378e13	−1.84155	−0.920774	+	0.390096i	$0.872442\pi$
$762$	0	0
$763$	−2.51973e13	−2.69149
$764$	0	0
$765$	1.90897e12	0.201522
$766$	0	0
$767$	−5.75127e11	−0.0600046
$768$	0	0
$769$	1.93798e12	0.199839	0.0999194	−	0.994996i	$-0.468142\pi$
$769$	1.93798e12	0.199839	0.0999194	+	0.994996i	$0.468142\pi$
$770$	0	0
$771$	−1.64328e13	−1.67481
$772$	0	0
$773$	2.48333e12	0.250166	0.125083	−	0.992146i	$-0.460080\pi$
$773$	2.48333e12	0.250166	0.125083	+	0.992146i	$0.460080\pi$
$774$	0	0
$775$	−1.65935e13	−1.65227
$776$	0	0
$777$	−1.31783e13	−1.29707
$778$	0	0
$779$	4.37993e12	0.426136
$780$	0	0
$781$	−3.41572e12	−0.328513
$782$	0	0
$783$	4.39977e12	0.418314
$784$	0	0
$785$	3.85128e12	0.361986
$786$	0	0
$787$	1.40039e13	1.30126	0.650628	−	0.759397i	$-0.274506\pi$
$787$	1.40039e13	1.30126	0.650628	+	0.759397i	$0.274506\pi$
$788$	0	0
$789$	−1.14256e13	−1.04962
$790$	0	0
$791$	−2.02861e13	−1.84249
$792$	0	0
$793$	3.15958e12	0.283726
$794$	0	0
$795$	9.75473e12	0.866090
$796$	0	0
$797$	1.88540e13	1.65516	0.827581	−	0.561346i	$-0.189716\pi$
$797$	1.88540e13	1.65516	0.827581	+	0.561346i	$0.189716\pi$
$798$	0	0
$799$	7.42734e12	0.644722
$800$	0	0
$801$	−2.51661e13	−2.16008
$802$	0	0
$803$	9.39196e11	0.0797143
$804$	0	0
$805$	1.74923e12	0.146813
$806$	0	0
$807$	−3.48890e12	−0.289573
$808$	0	0
$809$	−1.03147e13	−0.846618	−0.423309	−	0.905985i	$-0.639132\pi$
$809$	−1.03147e13	−0.846618	−0.423309	+	0.905985i	$0.639132\pi$
$810$	0	0
$811$	−1.52992e13	−1.24187	−0.620934	−	0.783863i	$-0.713246\pi$
$811$	−1.52992e13	−1.24187	−0.620934	+	0.783863i	$0.713246\pi$
$812$	0	0
$813$	−2.85025e13	−2.28810
$814$	0	0
$815$	5.59278e12	0.444036
$816$	0	0
$817$	−8.15685e12	−0.640506
$818$	0	0
$819$	−7.63907e12	−0.593285
$820$	0	0
$821$	1.13333e13	0.870584	0.435292	−	0.900289i	$-0.356645\pi$
$821$	1.13333e13	0.870584	0.435292	+	0.900289i	$0.356645\pi$
$822$	0	0
$823$	−1.11886e13	−0.850109	−0.425055	−	0.905168i	$-0.639745\pi$
$823$	−1.11886e13	−0.850109	−0.425055	+	0.905168i	$0.639745\pi$
$824$	0	0
$825$	8.47584e12	0.637001
$826$	0	0
$827$	2.42673e13	1.80404	0.902020	−	0.431694i	$-0.142084\pi$
$827$	2.42673e13	1.80404	0.902020	+	0.431694i	$0.142084\pi$
$828$	0	0
$829$	−1.05896e13	−0.778723	−0.389362	−	0.921085i	$-0.627304\pi$
$829$	−1.05896e13	−0.778723	−0.389362	+	0.921085i	$0.627304\pi$
$830$	0	0
$831$	2.19592e13	1.59739
$832$	0	0
$833$	1.29959e13	0.935196
$834$	0	0
$835$	1.39543e11	0.00993391
$836$	0	0
$837$	9.35825e12	0.659068
$838$	0	0
$839$	7.99276e12	0.556888	0.278444	−	0.960452i	$-0.410181\pi$
$839$	7.99276e12	0.556888	0.278444	+	0.960452i	$0.410181\pi$
$840$	0	0
$841$	6.20969e12	0.428044
$842$	0	0
$843$	−3.67103e13	−2.50359
$844$	0	0
$845$	3.98889e11	0.0269152
$846$	0	0
$847$	1.98390e13	1.32447
$848$	0	0
$849$	−1.61780e13	−1.06866
$850$	0	0
$851$	−1.85441e12	−0.121206
$852$	0	0
$853$	2.38359e13	1.54156	0.770782	−	0.637099i	$-0.219866\pi$
$853$	2.38359e13	1.54156	0.770782	+	0.637099i	$0.219866\pi$
$854$	0	0
$855$	4.72579e12	0.302432
$856$	0	0
$857$	8.25953e12	0.523048	0.261524	−	0.965197i	$-0.415775\pi$
$857$	8.25953e12	0.523048	0.261524	+	0.965197i	$0.415775\pi$
$858$	0	0
$859$	9.39817e11	0.0588944	0.0294472	−	0.999566i	$-0.490625\pi$
$859$	9.39817e11	0.0588944	0.0294472	+	0.999566i	$0.490625\pi$
$860$	0	0
$861$	−2.54197e13	−1.57637
$862$	0	0
$863$	2.34275e13	1.43773	0.718866	−	0.695149i	$-0.244661\pi$
$863$	2.34275e13	1.43773	0.718866	+	0.695149i	$0.244661\pi$
$864$	0	0
$865$	−2.52513e12	−0.153360
$866$	0	0
$867$	−1.94527e13	−1.16922
$868$	0	0
$869$	4.94033e12	0.293878
$870$	0	0
$871$	3.60392e12	0.212175
$872$	0	0
$873$	2.61452e13	1.52345
$874$	0	0
$875$	1.97436e13	1.13865
$876$	0	0
$877$	−7.28411e12	−0.415794	−0.207897	−	0.978151i	$-0.566662\pi$
$877$	−7.28411e12	−0.415794	−0.207897	+	0.978151i	$0.566662\pi$
$878$	0	0
$879$	4.04452e13	2.28516
$880$	0	0
$881$	2.69854e12	0.150917	0.0754583	−	0.997149i	$-0.475958\pi$
$881$	2.69854e12	0.150917	0.0754583	+	0.997149i	$0.475958\pi$
$882$	0	0
$883$	−2.16728e13	−1.19975	−0.599875	−	0.800094i	$-0.704783\pi$
$883$	−2.16728e13	−1.19975	−0.599875	+	0.800094i	$0.704783\pi$
$884$	0	0
$885$	−2.06491e12	−0.113151
$886$	0	0
$887$	2.91657e12	0.158203	0.0791017	−	0.996867i	$-0.474795\pi$
$887$	2.91657e12	0.158203	0.0791017	+	0.996867i	$0.474795\pi$
$888$	0	0
$889$	5.59048e13	3.00186
$890$	0	0
$891$	6.49522e12	0.345258
$892$	0	0
$893$	1.83870e13	0.967561
$894$	0	0
$895$	−3.73606e12	−0.194630
$896$	0	0
$897$	−1.94593e12	−0.100360
$898$	0	0
$899$	4.40644e13	2.24993
$900$	0	0
$901$	1.52870e13	0.772789
$902$	0	0
$903$	4.73398e13	2.36936
$904$	0	0
$905$	9.95751e12	0.493437
$906$	0	0
$907$	−4.15436e12	−0.203831	−0.101916	−	0.994793i	$-0.532497\pi$
$907$	−4.15436e12	−0.203831	−0.101916	+	0.994793i	$0.532497\pi$
$908$	0	0
$909$	1.09332e13	0.531141
$910$	0	0
$911$	1.49662e13	0.719911	0.359955	−	0.932970i	$-0.382792\pi$
$911$	1.49662e13	0.719911	0.359955	+	0.932970i	$0.382792\pi$
$912$	0	0
$913$	3.01167e12	0.143446
$914$	0	0
$915$	1.13440e13	0.535022
$916$	0	0
$917$	−2.64329e13	−1.23448
$918$	0	0
$919$	2.15787e13	0.997940	0.498970	−	0.866619i	$-0.333712\pi$
$919$	2.15787e13	0.997940	0.498970	+	0.866619i	$0.333712\pi$
$920$	0	0
$921$	7.34699e12	0.336466
$922$	0	0
$923$	4.13705e12	0.187622
$924$	0	0
$925$	−9.78300e12	−0.439374
$926$	0	0
$927$	−2.89119e13	−1.28593
$928$	0	0
$929$	1.19492e12	0.0526344	0.0263172	−	0.999654i	$-0.491622\pi$
$929$	1.19492e12	0.0526344	0.0263172	+	0.999654i	$0.491622\pi$
$930$	0	0
$931$	3.21723e13	1.40349
$932$	0	0
$933$	−6.51066e13	−2.81292
$934$	0	0
$935$	−1.85306e12	−0.0792933
$936$	0	0
$937$	−9.57530e12	−0.405811	−0.202906	−	0.979198i	$-0.565038\pi$
$937$	−9.57530e12	−0.405811	−0.202906	+	0.979198i	$0.565038\pi$
$938$	0	0
$939$	3.85218e13	1.61700
$940$	0	0
$941$	9.31828e12	0.387420	0.193710	−	0.981059i	$-0.437948\pi$
$941$	9.31828e12	0.387420	0.193710	+	0.981059i	$0.437948\pi$
$942$	0	0
$943$	−3.57700e12	−0.147305
$944$	0	0
$945$	−5.20436e12	−0.212287
$946$	0	0
$947$	1.73521e13	0.701095	0.350547	−	0.936545i	$-0.385996\pi$
$947$	1.73521e13	0.701095	0.350547	+	0.936545i	$0.385996\pi$
$948$	0	0
$949$	−1.13754e12	−0.0455268
$950$	0	0
$951$	−4.62532e13	−1.83371
$952$	0	0
$953$	−4.44704e13	−1.74644	−0.873219	−	0.487327i	$-0.837972\pi$
$953$	−4.44704e13	−1.74644	−0.873219	+	0.487327i	$0.837972\pi$
$954$	0	0
$955$	−1.07650e13	−0.418791
$956$	0	0
$957$	−2.25077e13	−0.867417
$958$	0	0
$959$	2.34614e13	0.895716
$960$	0	0
$961$	6.72847e13	2.54484
$962$	0	0
$963$	−2.16026e13	−0.809447
$964$	0	0
$965$	9.19028e12	0.341158
$966$	0	0
$967$	−4.23612e13	−1.55793	−0.778967	−	0.627064i	$-0.784256\pi$
$967$	−4.23612e13	−1.55793	−0.778967	+	0.627064i	$0.784256\pi$
$968$	0	0
$969$	1.34066e13	0.488498
$970$	0	0
$971$	−2.19787e13	−0.793443	−0.396722	−	0.917939i	$-0.629852\pi$
$971$	−2.19787e13	−0.793443	−0.396722	+	0.917939i	$0.629852\pi$
$972$	0	0
$973$	−9.87244e12	−0.353116
$974$	0	0
$975$	−1.02658e13	−0.363807
$976$	0	0
$977$	3.03573e13	1.06595	0.532976	−	0.846130i	$-0.321074\pi$
$977$	3.03573e13	1.06595	0.532976	+	0.846130i	$0.321074\pi$
$978$	0	0
$979$	2.44290e13	0.849931
$980$	0	0
$981$	5.55948e13	1.91657
$982$	0	0
$983$	5.32931e13	1.82046	0.910228	−	0.414108i	$-0.135907\pi$
$983$	5.32931e13	1.82046	0.910228	+	0.414108i	$0.135907\pi$
$984$	0	0
$985$	7.46004e12	0.252510
$986$	0	0
$987$	−1.06712e14	−3.57920
$988$	0	0
$989$	6.66153e12	0.221407
$990$	0	0
$991$	−5.41382e13	−1.78309	−0.891544	−	0.452935i	$-0.850377\pi$
$991$	−5.41382e13	−1.78309	−0.891544	+	0.452935i	$0.850377\pi$
$992$	0	0
$993$	9.46253e12	0.308841
$994$	0	0
$995$	−1.17926e13	−0.381422
$996$	0	0
$997$	−3.76703e13	−1.20746	−0.603728	−	0.797191i	$-0.706319\pi$
$997$	−3.76703e13	−1.20746	−0.603728	+	0.797191i	$0.706319\pi$
$998$	0	0
$999$	5.51731e12	0.175260

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	208.10.a.m.1.7		8
4.3	odd	2		104.10.a.d.1.2	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
104.10.a.d.1.2	✓	8	4.3	odd	2
208.10.a.m.1.7		8	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)

\(f(q)\)	\(=\)	\(q+209.704 q^{3} +488.996 q^{5} -11010.2 q^{7} +24292.6 q^{9} -23581.1 q^{11} +28561.0 q^{13} +102544. q^{15} +160701. q^{17} +397828. q^{19} -2.30887e6 q^{21} -324898. q^{23} -1.71401e6 q^{25} +966648. q^{27} +4.55158e6 q^{29} +9.68113e6 q^{31} -4.94504e6 q^{33} -5.38392e6 q^{35} +5.70767e6 q^{37} +5.98934e6 q^{39} +1.10096e7 q^{41} -2.05034e7 q^{43} +1.18790e7 q^{45} +4.62183e7 q^{47} +8.08698e7 q^{49} +3.36996e7 q^{51} +9.51270e7 q^{53} -1.15311e7 q^{55} +8.34260e7 q^{57} -2.01368e7 q^{59} +1.10626e8 q^{61} -2.67465e8 q^{63} +1.39662e7 q^{65} +1.26183e8 q^{67} -6.81323e7 q^{69} +1.44850e8 q^{71} -3.98283e7 q^{73} -3.59434e8 q^{75} +2.59632e8 q^{77} -2.09504e8 q^{79} -2.75441e8 q^{81} -1.27715e8 q^{83} +7.85822e7 q^{85} +9.54482e8 q^{87} -1.03596e9 q^{89} -3.14461e8 q^{91} +2.03017e9 q^{93} +1.94536e8 q^{95} +1.07626e9 q^{97} -5.72846e8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 141 q^{3} + 2051 q^{5} + 2417 q^{7} + 115741 q^{9} + 53118 q^{11} + 228488 q^{13} + 464555 q^{15} + 433095 q^{17} + 434954 q^{19} + 906875 q^{21} + 1124296 q^{23} + 5966065 q^{25} - 7820643 q^{27}+ \cdots + 641626736 q^{99}+O(q^{100}) \) 8 * q - 141 * q^3 + 2051 * q^5 + 2417 * q^7 + 115741 * q^9 + 53118 * q^11 + 228488 * q^13 + 464555 * q^15 + 433095 * q^17 + 434954 * q^19 + 906875 * q^21 + 1124296 * q^23 + 5966065 * q^25 - 7820643 * q^27 - 1913432 * q^29 - 5220148 * q^31 - 1626882 * q^33 - 950707 * q^35 - 419409 * q^37 - 4027101 * q^39 + 12148046 * q^41 - 32143187 * q^43 + 18893544 * q^45 - 57898411 * q^47 + 104302637 * q^49 - 198521477 * q^51 + 273106866 * q^53 - 141214662 * q^55 + 253032058 * q^57 - 50734502 * q^59 + 139520018 * q^61 + 20344158 * q^63 + 58578611 * q^65 - 528144786 * q^67 + 483506576 * q^69 - 334814973 * q^71 + 630864344 * q^73 - 2265796264 * q^75 + 939664146 * q^77 - 865037608 * q^79 + 3194281040 * q^81 - 1520229952 * q^83 + 2130158385 * q^85 - 1167867006 * q^87 + 894700240 * q^89 + 69031937 * q^91 + 3803209548 * q^93 + 2695926542 * q^95 + 980232164 * q^97 + 641626736 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more