LMFDB - Embedded newform 208.10.a.l.1.7

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			1.7
Root			$233.704$ 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)

$f(q)$	$=$	$q+231.704 q^{3} +1363.19 q^{5} -3005.73 q^{7} +34003.6 q^{9} -84978.4 q^{11} -28561.0 q^{13} +315857. q^{15} -358184. q^{17} -315187. q^{19} -696439. q^{21} -649646. q^{23} -94826.5 q^{25} +3.31814e6 q^{27} -4.28284e6 q^{29} -1.18999e6 q^{31} -1.96898e7 q^{33} -4.09739e6 q^{35} +5.51296e6 q^{37} -6.61769e6 q^{39} -257315. q^{41} +799842. q^{43} +4.63535e7 q^{45} -3.91606e7 q^{47} -3.13192e7 q^{49} -8.29926e7 q^{51} +3.04463e7 q^{53} -1.15842e8 q^{55} -7.30300e7 q^{57} -1.67474e8 q^{59} +6.85381e7 q^{61} -1.02206e8 q^{63} -3.89342e7 q^{65} +2.58838e8 q^{67} -1.50525e8 q^{69} -2.12909e8 q^{71} +3.19100e8 q^{73} -2.19717e7 q^{75} +2.55422e8 q^{77} +2.88383e8 q^{79} +9.95316e7 q^{81} +6.75948e8 q^{83} -4.88274e8 q^{85} -9.92350e8 q^{87} +1.13653e9 q^{89} +8.58467e7 q^{91} -2.75726e8 q^{93} -4.29661e8 q^{95} -3.65840e8 q^{97} -2.88957e9 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	231.704	1.65153	0.825767	−	0.564012i	$-0.190743\pi$
$3$	231.704	1.65153	0.825767	+	0.564012i	$0.190743\pi$
$4$	0	0
$5$	1363.19	0.975422	0.487711	−	0.873005i	$-0.337832\pi$
$5$	1363.19	0.975422	0.487711	+	0.873005i	$0.337832\pi$
$6$	0	0
$7$	−3005.73	−0.473161	−0.236580	−	0.971612i	$-0.576027\pi$
$7$	−3005.73	−0.473161	−0.236580	+	0.971612i	$0.576027\pi$
$8$	0	0
$9$	34003.6	1.72756
$10$	0	0
$11$	−84978.4	−1.75001	−0.875007	−	0.484111i	$-0.839143\pi$
$11$	−84978.4	−1.75001	−0.875007	+	0.484111i	$0.839143\pi$
$12$	0	0
$13$	−28561.0	−0.277350
$13$	−28561.0	−0.277350
$14$	0	0
$15$	315857.	1.61094
$16$	0	0
$17$	−358184.	−1.04013	−0.520063	−	0.854128i	$-0.674092\pi$
$17$	−358184.	−1.04013	−0.520063	+	0.854128i	$0.674092\pi$
$18$	0	0
$19$	−315187.	−0.554852	−0.277426	−	0.960747i	$-0.589481\pi$
$19$	−315187.	−0.554852	−0.277426	+	0.960747i	$0.589481\pi$
$20$	0	0
$21$	−696439.	−0.781441
$22$	0	0
$23$	−649646.	−0.484062	−0.242031	−	0.970268i	$-0.577814\pi$
$23$	−649646.	−0.484062	−0.242031	+	0.970268i	$0.577814\pi$
$24$	0	0
$25$	−94826.5	−0.0485512
$26$	0	0
$27$	3.31814e6	1.20159
$28$	0	0
$29$	−4.28284e6	−1.12445	−0.562226	−	0.826984i	$-0.690055\pi$
$29$	−4.28284e6	−1.12445	−0.562226	+	0.826984i	$0.690055\pi$
$30$	0	0
$31$	−1.18999e6	−0.231429	−0.115714	−	0.993283i	$-0.536916\pi$
$31$	−1.18999e6	−0.231429	−0.115714	+	0.993283i	$0.536916\pi$
$32$	0	0
$33$	−1.96898e7	−2.89021
$34$	0	0
$35$	−4.09739e6	−0.461532
$36$	0	0
$37$	5.51296e6	0.483590	0.241795	−	0.970327i	$-0.422264\pi$
$37$	5.51296e6	0.483590	0.241795	+	0.970327i	$0.422264\pi$
$38$	0	0
$39$	−6.61769e6	−0.458053
$40$	0	0
$41$	−257315.	−0.0142212	−0.00711062	−	0.999975i	$-0.502263\pi$
$41$	−257315.	−0.0142212	−0.00711062	+	0.999975i	$0.502263\pi$
$42$	0	0
$43$	799842.	0.0356777	0.0178388	−	0.999841i	$-0.494321\pi$
$43$	799842.	0.0356777	0.0178388	+	0.999841i	$0.494321\pi$
$44$	0	0
$45$	4.63535e7	1.68510
$46$	0	0
$47$	−3.91606e7	−1.17060	−0.585301	−	0.810816i	$-0.699024\pi$
$47$	−3.91606e7	−1.17060	−0.585301	+	0.810816i	$0.699024\pi$
$48$	0	0
$49$	−3.13192e7	−0.776119
$50$	0	0
$51$	−8.29926e7	−1.71780
$52$	0	0
$53$	3.04463e7	0.530021	0.265010	−	0.964246i	$-0.414625\pi$
$53$	3.04463e7	0.530021	0.265010	+	0.964246i	$0.414625\pi$
$54$	0	0
$55$	−1.15842e8	−1.70700
$56$	0	0
$57$	−7.30300e7	−0.916356
$58$	0	0
$59$	−1.67474e8	−1.79934	−0.899672	−	0.436566i	$-0.856194\pi$
$59$	−1.67474e8	−1.79934	−0.899672	+	0.436566i	$0.856194\pi$
$60$	0	0
$61$	6.85381e7	0.633793	0.316897	−	0.948460i	$-0.397359\pi$
$61$	6.85381e7	0.633793	0.316897	+	0.948460i	$0.397359\pi$
$62$	0	0
$63$	−1.02206e8	−0.817415
$64$	0	0
$65$	−3.89342e7	−0.270533
$66$	0	0
$67$	2.58838e8	1.56925	0.784624	−	0.619971i	$-0.212856\pi$
$67$	2.58838e8	1.56925	0.784624	+	0.619971i	$0.212856\pi$
$68$	0	0
$69$	−1.50525e8	−0.799445
$70$	0	0
$71$	−2.12909e8	−0.994333	−0.497167	−	0.867655i	$-0.665626\pi$
$71$	−2.12909e8	−0.994333	−0.497167	+	0.867655i	$0.665626\pi$
$72$	0	0
$73$	3.19100e8	1.31515	0.657573	−	0.753391i	$-0.271583\pi$
$73$	3.19100e8	1.31515	0.657573	+	0.753391i	$0.271583\pi$
$74$	0	0
$75$	−2.19717e7	−0.0801839
$76$	0	0
$77$	2.55422e8	0.828038
$78$	0	0
$79$	2.88383e8	0.833006	0.416503	−	0.909134i	$-0.363256\pi$
$79$	2.88383e8	0.833006	0.416503	+	0.909134i	$0.363256\pi$
$80$	0	0
$81$	9.95316e7	0.256908
$82$	0	0
$83$	6.75948e8	1.56337	0.781685	−	0.623674i	$-0.214361\pi$
$83$	6.75948e8	1.56337	0.781685	+	0.623674i	$0.214361\pi$
$84$	0	0
$85$	−4.88274e8	−1.01456
$86$	0	0
$87$	−9.92350e8	−1.85707
$88$	0	0
$89$	1.13653e9	1.92011	0.960055	−	0.279812i	$-0.0902722\pi$
$89$	1.13653e9	1.92011	0.960055	+	0.279812i	$0.0902722\pi$
$90$	0	0
$91$	8.58467e7	0.131231
$92$	0	0
$93$	−2.75726e8	−0.382212
$94$	0	0
$95$	−4.29661e8	−0.541215
$96$	0	0
$97$	−3.65840e8	−0.419583	−0.209792	−	0.977746i	$-0.567279\pi$
$97$	−3.65840e8	−0.419583	−0.209792	+	0.977746i	$0.567279\pi$
$98$	0	0
$99$	−2.88957e9	−3.02326
$100$	0	0
$101$	−1.15655e9	−1.10590	−0.552952	−	0.833213i	$-0.686499\pi$
$101$	−1.15655e9	−1.10590	−0.552952	+	0.833213i	$0.686499\pi$
$102$	0	0
$103$	2.27280e8	0.198973	0.0994865	−	0.995039i	$-0.468280\pi$
$103$	2.27280e8	0.198973	0.0994865	+	0.995039i	$0.468280\pi$
$104$	0	0
$105$	−9.49381e8	−0.762235
$106$	0	0
$107$	−9.67474e8	−0.713530	−0.356765	−	0.934194i	$-0.616120\pi$
$107$	−9.67474e8	−0.713530	−0.356765	+	0.934194i	$0.616120\pi$
$108$	0	0
$109$	2.60415e9	1.76704	0.883520	−	0.468394i	$-0.155167\pi$
$109$	2.60415e9	1.76704	0.883520	+	0.468394i	$0.155167\pi$
$110$	0	0
$111$	1.27737e9	0.798664
$112$	0	0
$113$	2.19137e9	1.26433	0.632167	−	0.774832i	$-0.282166\pi$
$113$	2.19137e9	1.26433	0.632167	+	0.774832i	$0.282166\pi$
$114$	0	0
$115$	−8.85593e8	−0.472165
$116$	0	0
$117$	−9.71177e8	−0.479139
$118$	0	0
$119$	1.07660e9	0.492147
$120$	0	0
$121$	4.86338e9	2.06255
$122$	0	0
$123$	−5.96208e7	−0.0234868
$124$	0	0
$125$	−2.79176e9	−1.02278
$126$	0	0
$127$	2.18989e9	0.746973	0.373487	−	0.927636i	$-0.378162\pi$
$127$	2.18989e9	0.746973	0.373487	+	0.927636i	$0.378162\pi$
$128$	0	0
$129$	1.85326e8	0.0589228
$130$	0	0
$131$	−1.90918e9	−0.566403	−0.283201	−	0.959060i	$-0.591397\pi$
$131$	−1.90918e9	−0.566403	−0.283201	+	0.959060i	$0.591397\pi$
$132$	0	0
$133$	9.47367e8	0.262534
$134$	0	0
$135$	4.52326e9	1.17206
$136$	0	0
$137$	−1.19958e9	−0.290930	−0.145465	−	0.989363i	$-0.546468\pi$
$137$	−1.19958e9	−0.290930	−0.145465	+	0.989363i	$0.546468\pi$
$138$	0	0
$139$	1.40633e9	0.319538	0.159769	−	0.987154i	$-0.448925\pi$
$139$	1.40633e9	0.319538	0.159769	+	0.987154i	$0.448925\pi$
$140$	0	0
$141$	−9.07366e9	−1.93329
$142$	0	0
$143$	2.42707e9	0.485366
$144$	0	0
$145$	−5.83834e9	−1.09682
$146$	0	0
$147$	−7.25677e9	−1.28179
$148$	0	0
$149$	−1.80364e9	−0.299786	−0.149893	−	0.988702i	$-0.547893\pi$
$149$	−1.80364e9	−0.299786	−0.149893	+	0.988702i	$0.547893\pi$
$150$	0	0
$151$	−1.05410e10	−1.65000	−0.825000	−	0.565132i	$-0.808825\pi$
$151$	−1.05410e10	−1.65000	−0.825000	+	0.565132i	$0.808825\pi$
$152$	0	0
$153$	−1.21795e10	−1.79688
$154$	0	0
$155$	−1.62219e9	−0.225741
$156$	0	0
$157$	−1.02542e10	−1.34696	−0.673479	−	0.739206i	$-0.735201\pi$
$157$	−1.02542e10	−1.34696	−0.673479	+	0.739206i	$0.735201\pi$
$158$	0	0
$159$	7.05451e9	0.875346
$160$	0	0
$161$	1.95266e9	0.229039
$162$	0	0
$163$	4.42748e8	0.0491261	0.0245630	−	0.999698i	$-0.492181\pi$
$163$	4.42748e8	0.0491261	0.0245630	+	0.999698i	$0.492181\pi$
$164$	0	0
$165$	−2.68410e10	−2.81917
$166$	0	0
$167$	−2.24534e9	−0.223387	−0.111694	−	0.993743i	$-0.535628\pi$
$167$	−2.24534e9	−0.223387	−0.111694	+	0.993743i	$0.535628\pi$
$168$	0	0
$169$	8.15731e8	0.0769231
$170$	0	0
$171$	−1.07175e10	−0.958541
$172$	0	0
$173$	3.81893e9	0.324141	0.162071	−	0.986779i	$-0.448183\pi$
$173$	3.81893e9	0.324141	0.162071	+	0.986779i	$0.448183\pi$
$174$	0	0
$175$	2.85023e8	0.0229725
$176$	0	0
$177$	−3.88044e10	−2.97168
$178$	0	0
$179$	−2.65522e10	−1.93313	−0.966567	−	0.256414i	$-0.917459\pi$
$179$	−2.65522e10	−1.93313	−0.966567	+	0.256414i	$0.917459\pi$
$180$	0	0
$181$	−1.23202e10	−0.853224	−0.426612	−	0.904435i	$-0.640293\pi$
$181$	−1.23202e10	−0.853224	−0.426612	+	0.904435i	$0.640293\pi$
$182$	0	0
$183$	1.58805e10	1.04673
$184$	0	0
$185$	7.51523e9	0.471704
$186$	0	0
$187$	3.04379e10	1.82024
$188$	0	0
$189$	−9.97342e9	−0.568547
$190$	0	0
$191$	−3.39753e10	−1.84720	−0.923598	−	0.383362i	$-0.874766\pi$
$191$	−3.39753e10	−1.84720	−0.923598	+	0.383362i	$0.874766\pi$
$192$	0	0
$193$	1.42354e10	0.738518	0.369259	−	0.929327i	$-0.379612\pi$
$193$	1.42354e10	0.738518	0.369259	+	0.929327i	$0.379612\pi$
$194$	0	0
$195$	−9.02120e9	−0.446795
$196$	0	0
$197$	3.21189e10	1.51937	0.759684	−	0.650292i	$-0.225353\pi$
$197$	3.21189e10	1.51937	0.759684	+	0.650292i	$0.225353\pi$
$198$	0	0
$199$	−3.30486e10	−1.49387	−0.746937	−	0.664895i	$-0.768476\pi$
$199$	−3.30486e10	−1.49387	−0.746937	+	0.664895i	$0.768476\pi$
$200$	0	0
$201$	5.99738e10	2.59167
$202$	0	0
$203$	1.28731e10	0.532047
$204$	0	0
$205$	−3.50770e8	−0.0138717
$206$	0	0
$207$	−2.20903e10	−0.836248
$208$	0	0
$209$	2.67841e10	0.970998
$210$	0	0
$211$	−3.46440e10	−1.20325	−0.601626	−	0.798778i	$-0.705480\pi$
$211$	−3.46440e10	−1.20325	−0.601626	+	0.798778i	$0.705480\pi$
$212$	0	0
$213$	−4.93319e10	−1.64217
$214$	0	0
$215$	1.09034e9	0.0348008
$216$	0	0
$217$	3.57680e9	0.109503
$218$	0	0
$219$	7.39366e10	2.17201
$220$	0	0
$221$	1.02301e10	0.288479
$222$	0	0
$223$	−1.05085e10	−0.284557	−0.142278	−	0.989827i	$-0.545443\pi$
$223$	−1.05085e10	−0.284557	−0.142278	+	0.989827i	$0.545443\pi$
$224$	0	0
$225$	−3.22444e9	−0.0838752
$226$	0	0
$227$	5.02538e10	1.25618	0.628090	−	0.778140i	$-0.283837\pi$
$227$	5.02538e10	1.25618	0.628090	+	0.778140i	$0.283837\pi$
$228$	0	0
$229$	−8.14531e10	−1.95726	−0.978629	−	0.205634i	$-0.934074\pi$
$229$	−8.14531e10	−1.95726	−0.978629	+	0.205634i	$0.934074\pi$
$230$	0	0
$231$	5.91822e10	1.36753
$232$	0	0
$233$	−4.16919e10	−0.926724	−0.463362	−	0.886169i	$-0.653357\pi$
$233$	−4.16919e10	−0.926724	−0.463362	+	0.886169i	$0.653357\pi$
$234$	0	0
$235$	−5.33835e10	−1.14183
$236$	0	0
$237$	6.68194e10	1.37574
$238$	0	0
$239$	2.59529e10	0.514512	0.257256	−	0.966343i	$-0.417182\pi$
$239$	2.59529e10	0.514512	0.257256	+	0.966343i	$0.417182\pi$
$240$	0	0
$241$	2.58108e10	0.492862	0.246431	−	0.969160i	$-0.420742\pi$
$241$	2.58108e10	0.492862	0.246431	+	0.969160i	$0.420742\pi$
$242$	0	0
$243$	−4.22490e10	−0.777300
$244$	0	0
$245$	−4.26941e10	−0.757044
$246$	0	0
$247$	9.00205e9	0.153888
$248$	0	0
$249$	1.56620e11	2.58196
$250$	0	0
$251$	4.41867e10	0.702684	0.351342	−	0.936247i	$-0.385725\pi$
$251$	4.41867e10	0.702684	0.351342	+	0.936247i	$0.385725\pi$
$252$	0	0
$253$	5.52059e10	0.847116
$254$	0	0
$255$	−1.13135e11	−1.67558
$256$	0	0
$257$	1.28156e11	1.83249	0.916244	−	0.400621i	$-0.131205\pi$
$257$	1.28156e11	1.83249	0.916244	+	0.400621i	$0.131205\pi$
$258$	0	0
$259$	−1.65705e10	−0.228816
$260$	0	0
$261$	−1.45632e11	−1.94256
$262$	0	0
$263$	−5.05576e10	−0.651607	−0.325804	−	0.945437i	$-0.605635\pi$
$263$	−5.05576e10	−0.651607	−0.325804	+	0.945437i	$0.605635\pi$
$264$	0	0
$265$	4.15042e10	0.516994
$266$	0	0
$267$	2.63338e11	3.17112
$268$	0	0
$269$	8.08338e10	0.941256	0.470628	−	0.882332i	$-0.344027\pi$
$269$	8.08338e10	0.941256	0.470628	+	0.882332i	$0.344027\pi$
$270$	0	0
$271$	−3.22924e10	−0.363696	−0.181848	−	0.983327i	$-0.558208\pi$
$271$	−3.22924e10	−0.363696	−0.181848	+	0.983327i	$0.558208\pi$
$272$	0	0
$273$	1.98910e10	0.216733
$274$	0	0
$275$	8.05821e9	0.0849652
$276$	0	0
$277$	9.35939e10	0.955187	0.477594	−	0.878581i	$-0.341509\pi$
$277$	9.35939e10	0.955187	0.477594	+	0.878581i	$0.341509\pi$
$278$	0	0
$279$	−4.04641e10	−0.399807
$280$	0	0
$281$	1.19578e11	1.14413	0.572063	−	0.820210i	$-0.306143\pi$
$281$	1.19578e11	1.14413	0.572063	+	0.820210i	$0.306143\pi$
$282$	0	0
$283$	−1.82984e11	−1.69580	−0.847899	−	0.530157i	$-0.822133\pi$
$283$	−1.82984e11	−1.69580	−0.847899	+	0.530157i	$0.822133\pi$
$284$	0	0
$285$	−9.95540e10	−0.893834
$286$	0	0
$287$	7.73419e8	0.00672893
$288$	0	0
$289$	9.70793e9	0.0818628
$290$	0	0
$291$	−8.47664e10	−0.692956
$292$	0	0
$293$	−1.56706e11	−1.24217	−0.621083	−	0.783744i	$-0.713307\pi$
$293$	−1.56706e11	−1.24217	−0.621083	+	0.783744i	$0.713307\pi$
$294$	0	0
$295$	−2.28300e11	−1.75512
$296$	0	0
$297$	−2.81970e11	−2.10280
$298$	0	0
$299$	1.85545e10	0.134255
$300$	0	0
$301$	−2.40411e9	−0.0168813
$302$	0	0
$303$	−2.67976e11	−1.82644
$304$	0	0
$305$	9.34307e10	0.618216
$306$	0	0
$307$	1.70113e10	0.109299	0.0546494	−	0.998506i	$-0.482596\pi$
$307$	1.70113e10	0.109299	0.0546494	+	0.998506i	$0.482596\pi$
$308$	0	0
$309$	5.26617e10	0.328611
$310$	0	0
$311$	2.20604e11	1.33719	0.668594	−	0.743628i	$-0.266896\pi$
$311$	2.20604e11	1.33719	0.668594	+	0.743628i	$0.266896\pi$
$312$	0	0
$313$	1.72838e11	1.01786	0.508931	−	0.860807i	$-0.330041\pi$
$313$	1.72838e11	1.01786	0.508931	+	0.860807i	$0.330041\pi$
$314$	0	0
$315$	−1.39326e11	−0.797325
$316$	0	0
$317$	−1.87242e11	−1.04144	−0.520722	−	0.853726i	$-0.674337\pi$
$317$	−1.87242e11	−1.04144	−0.520722	+	0.853726i	$0.674337\pi$
$318$	0	0
$319$	3.63949e11	1.96781
$320$	0	0
$321$	−2.24167e11	−1.17842
$322$	0	0
$323$	1.12895e11	0.577116
$324$	0	0
$325$	2.70834e9	0.0134657
$326$	0	0
$327$	6.03391e11	2.91832
$328$	0	0
$329$	1.17706e11	0.553883
$330$	0	0
$331$	−7.24889e10	−0.331929	−0.165965	−	0.986132i	$-0.553074\pi$
$331$	−7.24889e10	−0.331929	−0.165965	+	0.986132i	$0.553074\pi$
$332$	0	0
$333$	1.87460e11	0.835431
$334$	0	0
$335$	3.52847e11	1.53068
$336$	0	0
$337$	−1.67327e11	−0.706693	−0.353347	−	0.935492i	$-0.614956\pi$
$337$	−1.67327e11	−0.706693	−0.353347	+	0.935492i	$0.614956\pi$
$338$	0	0
$339$	5.07748e11	2.08809
$340$	0	0
$341$	1.01124e11	0.405003
$342$	0	0
$343$	2.15429e11	0.840390
$344$	0	0
$345$	−2.05195e11	−0.779797
$346$	0	0
$347$	−1.33712e11	−0.495095	−0.247548	−	0.968876i	$-0.579625\pi$
$347$	−1.33712e11	−0.495095	−0.247548	+	0.968876i	$0.579625\pi$
$348$	0	0
$349$	−2.28882e10	−0.0825842	−0.0412921	−	0.999147i	$-0.513147\pi$
$349$	−2.28882e10	−0.0825842	−0.0412921	+	0.999147i	$0.513147\pi$
$350$	0	0
$351$	−9.47693e10	−0.333262
$352$	0	0
$353$	−1.87200e11	−0.641680	−0.320840	−	0.947133i	$-0.603965\pi$
$353$	−1.87200e11	−0.641680	−0.320840	+	0.947133i	$0.603965\pi$
$354$	0	0
$355$	−2.90237e11	−0.969895
$356$	0	0
$357$	2.49453e11	0.812797
$358$	0	0
$359$	2.70226e11	0.858621	0.429311	−	0.903157i	$-0.358757\pi$
$359$	2.70226e11	0.858621	0.429311	+	0.903157i	$0.358757\pi$
$360$	0	0
$361$	−2.23345e11	−0.692140
$362$	0	0
$363$	1.12686e12	3.40637
$364$	0	0
$365$	4.34995e11	1.28282
$366$	0	0
$367$	−1.23357e11	−0.354949	−0.177474	−	0.984125i	$-0.556793\pi$
$367$	−1.23357e11	−0.354949	−0.177474	+	0.984125i	$0.556793\pi$
$368$	0	0
$369$	−8.74963e9	−0.0245681
$370$	0	0
$371$	−9.15133e10	−0.250785
$372$	0	0
$373$	3.23044e11	0.864116	0.432058	−	0.901846i	$-0.357788\pi$
$373$	3.23044e11	0.864116	0.432058	+	0.901846i	$0.357788\pi$
$374$	0	0
$375$	−6.46860e11	−1.68916
$376$	0	0
$377$	1.22322e11	0.311867
$378$	0	0
$379$	1.68579e11	0.419687	0.209844	−	0.977735i	$-0.432705\pi$
$379$	1.68579e11	0.419687	0.209844	+	0.977735i	$0.432705\pi$
$380$	0	0
$381$	5.07405e11	1.23365
$382$	0	0
$383$	−7.88973e11	−1.87356	−0.936780	−	0.349918i	$-0.886209\pi$
$383$	−7.88973e11	−1.87356	−0.936780	+	0.349918i	$0.886209\pi$
$384$	0	0
$385$	3.48190e11	0.807687
$386$	0	0
$387$	2.71975e10	0.0616354
$388$	0	0
$389$	−1.33409e11	−0.295400	−0.147700	−	0.989032i	$-0.547187\pi$
$389$	−1.33409e11	−0.295400	−0.147700	+	0.989032i	$0.547187\pi$
$390$	0	0
$391$	2.32693e11	0.503486
$392$	0	0
$393$	−4.42363e11	−0.935433
$394$	0	0
$395$	3.93122e11	0.812532
$396$	0	0
$397$	−3.70840e11	−0.749255	−0.374628	−	0.927175i	$-0.622229\pi$
$397$	−3.70840e11	−0.749255	−0.374628	+	0.927175i	$0.622229\pi$
$398$	0	0
$399$	2.19508e11	0.433584
$400$	0	0
$401$	−4.67514e11	−0.902912	−0.451456	−	0.892293i	$-0.649095\pi$
$401$	−4.67514e11	−0.902912	−0.451456	+	0.892293i	$0.649095\pi$
$402$	0	0
$403$	3.39874e10	0.0641868
$404$	0	0
$405$	1.35681e11	0.250594
$406$	0	0
$407$	−4.68482e11	−0.846289
$408$	0	0
$409$	7.00780e11	1.23830	0.619151	−	0.785272i	$-0.287477\pi$
$409$	7.00780e11	1.23830	0.619151	+	0.785272i	$0.287477\pi$
$410$	0	0
$411$	−2.77948e11	−0.480480
$412$	0	0
$413$	5.03383e11	0.851379
$414$	0	0
$415$	9.21448e11	1.52495
$416$	0	0
$417$	3.25853e11	0.527727
$418$	0	0
$419$	9.60695e11	1.52273	0.761364	−	0.648324i	$-0.224530\pi$
$419$	9.60695e11	1.52273	0.761364	+	0.648324i	$0.224530\pi$
$420$	0	0
$421$	1.82836e11	0.283657	0.141828	−	0.989891i	$-0.454702\pi$
$421$	1.82836e11	0.283657	0.141828	+	0.989891i	$0.454702\pi$
$422$	0	0
$423$	−1.33160e12	−2.02229
$424$	0	0
$425$	3.39653e10	0.0504994
$426$	0	0
$427$	−2.06007e11	−0.299886
$428$	0	0
$429$	5.62361e11	0.801599
$430$	0	0
$431$	−7.99636e11	−1.11621	−0.558103	−	0.829772i	$-0.688471\pi$
$431$	−7.99636e11	−1.11621	−0.558103	+	0.829772i	$0.688471\pi$
$432$	0	0
$433$	−7.10514e11	−0.971353	−0.485676	−	0.874139i	$-0.661427\pi$
$433$	−7.10514e11	−0.971353	−0.485676	+	0.874139i	$0.661427\pi$
$434$	0	0
$435$	−1.35277e12	−1.81143
$436$	0	0
$437$	2.04760e11	0.268583
$438$	0	0
$439$	−2.00643e11	−0.257830	−0.128915	−	0.991656i	$-0.541149\pi$
$439$	−2.00643e11	−0.257830	−0.128915	+	0.991656i	$0.541149\pi$
$440$	0	0
$441$	−1.06497e12	−1.34079
$442$	0	0
$443$	−6.10644e11	−0.753305	−0.376653	−	0.926355i	$-0.622925\pi$
$443$	−6.10644e11	−0.753305	−0.376653	+	0.926355i	$0.622925\pi$
$444$	0	0
$445$	1.54931e12	1.87292
$446$	0	0
$447$	−4.17910e11	−0.495107
$448$	0	0
$449$	−2.36542e11	−0.274663	−0.137331	−	0.990525i	$-0.543853\pi$
$449$	−2.36542e11	−0.274663	−0.137331	+	0.990525i	$0.543853\pi$
$450$	0	0
$451$	2.18662e10	0.0248874
$452$	0	0
$453$	−2.44238e12	−2.72503
$454$	0	0
$455$	1.17026e11	0.128006
$456$	0	0
$457$	−9.93183e10	−0.106514	−0.0532570	−	0.998581i	$-0.516960\pi$
$457$	−9.93183e10	−0.106514	−0.0532570	+	0.998581i	$0.516960\pi$
$458$	0	0
$459$	−1.18850e12	−1.24981
$460$	0	0
$461$	−8.62553e11	−0.889471	−0.444735	−	0.895662i	$-0.646702\pi$
$461$	−8.62553e11	−0.889471	−0.444735	+	0.895662i	$0.646702\pi$
$462$	0	0
$463$	5.44955e11	0.551120	0.275560	−	0.961284i	$-0.411137\pi$
$463$	5.44955e11	0.551120	0.275560	+	0.961284i	$0.411137\pi$
$464$	0	0
$465$	−3.75868e11	−0.372818
$466$	0	0
$467$	1.17483e12	1.14300	0.571502	−	0.820601i	$-0.306361\pi$
$467$	1.17483e12	1.14300	0.571502	+	0.820601i	$0.306361\pi$
$468$	0	0
$469$	−7.77998e11	−0.742507
$470$	0	0
$471$	−2.37594e12	−2.22455
$472$	0	0
$473$	−6.79693e10	−0.0624364
$474$	0	0
$475$	2.98881e10	0.0269387
$476$	0	0
$477$	1.03528e12	0.915643
$478$	0	0
$479$	1.83595e12	1.59350	0.796748	−	0.604311i	$-0.206552\pi$
$479$	1.83595e12	1.59350	0.796748	+	0.604311i	$0.206552\pi$
$480$	0	0
$481$	−1.57456e11	−0.134124
$482$	0	0
$483$	4.52438e11	0.378266
$484$	0	0
$485$	−4.98711e11	−0.409271
$486$	0	0
$487$	4.49688e11	0.362269	0.181134	−	0.983458i	$-0.442023\pi$
$487$	4.49688e11	0.362269	0.181134	+	0.983458i	$0.442023\pi$
$488$	0	0
$489$	1.02586e11	0.0811334
$490$	0	0
$491$	−1.13272e12	−0.879538	−0.439769	−	0.898111i	$-0.644940\pi$
$491$	−1.13272e12	−0.879538	−0.439769	+	0.898111i	$0.644940\pi$
$492$	0	0
$493$	1.53405e12	1.16957
$494$	0	0
$495$	−3.93905e12	−2.94895
$496$	0	0
$497$	6.39948e11	0.470480
$498$	0	0
$499$	−2.34370e11	−0.169220	−0.0846098	−	0.996414i	$-0.526964\pi$
$499$	−2.34370e11	−0.169220	−0.0846098	+	0.996414i	$0.526964\pi$
$500$	0	0
$501$	−5.20254e11	−0.368931
$502$	0	0
$503$	−9.14574e11	−0.637035	−0.318517	−	0.947917i	$-0.603185\pi$
$503$	−9.14574e11	−0.637035	−0.318517	+	0.947917i	$0.603185\pi$
$504$	0	0
$505$	−1.57660e12	−1.07872
$506$	0	0
$507$	1.89008e11	0.127041
$508$	0	0
$509$	−2.69648e12	−1.78060	−0.890301	−	0.455373i	$-0.849506\pi$
$509$	−2.69648e12	−1.78060	−0.890301	+	0.455373i	$0.849506\pi$
$510$	0	0
$511$	−9.59128e11	−0.622276
$512$	0	0
$513$	−1.04583e12	−0.666706
$514$	0	0
$515$	3.09827e11	0.194083
$516$	0	0
$517$	3.32781e12	2.04857
$518$	0	0
$519$	8.84861e11	0.535330
$520$	0	0
$521$	2.56880e11	0.152742	0.0763712	−	0.997079i	$-0.475667\pi$
$521$	2.56880e11	0.152742	0.0763712	+	0.997079i	$0.475667\pi$
$522$	0	0
$523$	2.91617e12	1.70434	0.852169	−	0.523266i	$-0.175287\pi$
$523$	2.91617e12	1.70434	0.852169	+	0.523266i	$0.175287\pi$
$524$	0	0
$525$	6.60409e10	0.0379399
$526$	0	0
$527$	4.26237e11	0.240715
$528$	0	0
$529$	−1.37911e12	−0.765684
$530$	0	0
$531$	−5.69473e12	−3.10848
$532$	0	0
$533$	7.34917e9	0.00394426
$534$	0	0
$535$	−1.31886e12	−0.695993
$536$	0	0
$537$	−6.15224e12	−3.19263
$538$	0	0
$539$	2.66145e12	1.35822
$540$	0	0
$541$	2.40787e12	1.20850	0.604248	−	0.796797i	$-0.293474\pi$
$541$	2.40787e12	1.20850	0.604248	+	0.796797i	$0.293474\pi$
$542$	0	0
$543$	−2.85463e12	−1.40913
$544$	0	0
$545$	3.54996e12	1.72361
$546$	0	0
$547$	1.80446e12	0.861794	0.430897	−	0.902401i	$-0.358197\pi$
$547$	1.80446e12	0.861794	0.430897	+	0.902401i	$0.358197\pi$
$548$	0	0
$549$	2.33054e12	1.09492
$550$	0	0
$551$	1.34990e12	0.623904
$552$	0	0
$553$	−8.66802e11	−0.394146
$554$	0	0
$555$	1.74131e12	0.779035
$556$	0	0
$557$	−3.09918e12	−1.36426	−0.682131	−	0.731230i	$-0.738947\pi$
$557$	−3.09918e12	−1.36426	−0.682131	+	0.731230i	$0.738947\pi$
$558$	0	0
$559$	−2.28443e10	−0.00989520
$560$	0	0
$561$	7.05258e12	3.00618
$562$	0	0
$563$	1.15491e12	0.484465	0.242232	−	0.970218i	$-0.422120\pi$
$563$	1.15491e12	0.484465	0.242232	+	0.970218i	$0.422120\pi$
$564$	0	0
$565$	2.98726e12	1.23326
$566$	0	0
$567$	−2.99165e11	−0.121559
$568$	0	0
$569$	−2.85199e12	−1.14062	−0.570311	−	0.821429i	$-0.693177\pi$
$569$	−2.85199e12	−1.14062	−0.570311	+	0.821429i	$0.693177\pi$
$570$	0	0
$571$	−1.27646e12	−0.502512	−0.251256	−	0.967921i	$-0.580844\pi$
$571$	−1.27646e12	−0.502512	−0.251256	+	0.967921i	$0.580844\pi$
$572$	0	0
$573$	−7.87220e12	−3.05071
$574$	0	0
$575$	6.16036e10	0.0235018
$576$	0	0
$577$	1.87473e11	0.0704120	0.0352060	−	0.999380i	$-0.488791\pi$
$577$	1.87473e11	0.0704120	0.0352060	+	0.999380i	$0.488791\pi$
$578$	0	0
$579$	3.29839e12	1.21969
$580$	0	0
$581$	−2.03172e12	−0.739725
$582$	0	0
$583$	−2.58728e12	−0.927543
$584$	0	0
$585$	−1.32390e12	−0.467363
$586$	0	0
$587$	5.26467e12	1.83021	0.915103	−	0.403221i	$-0.132109\pi$
$587$	5.26467e12	1.83021	0.915103	+	0.403221i	$0.132109\pi$
$588$	0	0
$589$	3.75071e11	0.128409
$590$	0	0
$591$	7.44208e12	2.50929
$592$	0	0
$593$	2.68817e12	0.892710	0.446355	−	0.894856i	$-0.352722\pi$
$593$	2.68817e12	0.892710	0.446355	+	0.894856i	$0.352722\pi$
$594$	0	0
$595$	1.46762e12	0.480051
$596$	0	0
$597$	−7.65748e12	−2.46718
$598$	0	0
$599$	−2.11643e12	−0.671713	−0.335857	−	0.941913i	$-0.609026\pi$
$599$	−2.11643e12	−0.671713	−0.335857	+	0.941913i	$0.609026\pi$
$600$	0	0
$601$	7.22761e11	0.225975	0.112987	−	0.993596i	$-0.463958\pi$
$601$	7.22761e11	0.225975	0.112987	+	0.993596i	$0.463958\pi$
$602$	0	0
$603$	8.80143e12	2.71097
$604$	0	0
$605$	6.62973e12	2.01186
$606$	0	0
$607$	−3.09373e12	−0.924982	−0.462491	−	0.886624i	$-0.653044\pi$
$607$	−3.09373e12	−0.924982	−0.462491	+	0.886624i	$0.653044\pi$
$608$	0	0
$609$	2.98274e12	0.878693
$610$	0	0
$611$	1.11847e12	0.324666
$612$	0	0
$613$	−1.72186e12	−0.492522	−0.246261	−	0.969204i	$-0.579202\pi$
$613$	−1.72186e12	−0.492522	−0.246261	+	0.969204i	$0.579202\pi$
$614$	0	0
$615$	−8.12747e10	−0.0229096
$616$	0	0
$617$	2.06830e12	0.574554	0.287277	−	0.957848i	$-0.407250\pi$
$617$	2.06830e12	0.574554	0.287277	+	0.957848i	$0.407250\pi$
$618$	0	0
$619$	3.31744e12	0.908228	0.454114	−	0.890944i	$-0.349956\pi$
$619$	3.31744e12	0.908228	0.454114	+	0.890944i	$0.349956\pi$
$620$	0	0
$621$	−2.15561e12	−0.581646
$622$	0	0
$623$	−3.41610e12	−0.908521
$624$	0	0
$625$	−3.62050e12	−0.949092
$626$	0	0
$627$	6.20597e12	1.60364
$628$	0	0
$629$	−1.97465e12	−0.502994
$630$	0	0
$631$	−3.59724e12	−0.903312	−0.451656	−	0.892192i	$-0.649167\pi$
$631$	−3.59724e12	−0.903312	−0.451656	+	0.892192i	$0.649167\pi$
$632$	0	0
$633$	−8.02714e12	−1.98721
$634$	0	0
$635$	2.98524e12	0.728614
$636$	0	0
$637$	8.94507e11	0.215257
$638$	0	0
$639$	−7.23968e12	−1.71777
$640$	0	0
$641$	3.01339e12	0.705008	0.352504	−	0.935810i	$-0.385330\pi$
$641$	3.01339e12	0.705008	0.352504	+	0.935810i	$0.385330\pi$
$642$	0	0
$643$	−6.13019e12	−1.41424	−0.707122	−	0.707091i	$-0.750007\pi$
$643$	−6.13019e12	−1.41424	−0.707122	+	0.707091i	$0.750007\pi$
$644$	0	0
$645$	2.52636e11	0.0574746
$646$	0	0
$647$	8.79418e12	1.97300	0.986498	−	0.163773i	$-0.0523663\pi$
$647$	8.79418e12	1.97300	0.986498	+	0.163773i	$0.0523663\pi$
$648$	0	0
$649$	1.42317e13	3.14888
$650$	0	0
$651$	8.28758e11	0.180848
$652$	0	0
$653$	−7.93494e12	−1.70779	−0.853895	−	0.520446i	$-0.825766\pi$
$653$	−7.93494e12	−1.70779	−0.853895	+	0.520446i	$0.825766\pi$
$654$	0	0
$655$	−2.60258e12	−0.552482
$656$	0	0
$657$	1.08505e13	2.27200
$658$	0	0
$659$	−3.99324e12	−0.824787	−0.412393	−	0.911006i	$-0.635307\pi$
$659$	−3.99324e12	−0.824787	−0.412393	+	0.911006i	$0.635307\pi$
$660$	0	0
$661$	6.52485e12	1.32942	0.664712	−	0.747099i	$-0.268554\pi$
$661$	6.52485e12	1.32942	0.664712	+	0.747099i	$0.268554\pi$
$662$	0	0
$663$	2.37035e12	0.476433
$664$	0	0
$665$	1.29144e12	0.256082
$666$	0	0
$667$	2.78233e12	0.544305
$668$	0	0
$669$	−2.43486e12	−0.469955
$670$	0	0
$671$	−5.82426e12	−1.10915
$672$	0	0
$673$	8.79096e12	1.65184	0.825921	−	0.563786i	$-0.190656\pi$
$673$	8.79096e12	1.65184	0.825921	+	0.563786i	$0.190656\pi$
$674$	0	0
$675$	−3.14647e11	−0.0583387
$676$	0	0
$677$	−1.01870e13	−1.86379	−0.931895	−	0.362729i	$-0.881845\pi$
$677$	−1.01870e13	−1.86379	−0.931895	+	0.362729i	$0.881845\pi$
$678$	0	0
$679$	1.09962e12	0.198530
$680$	0	0
$681$	1.16440e13	2.07462
$682$	0	0
$683$	−6.13932e11	−0.107951	−0.0539756	−	0.998542i	$-0.517189\pi$
$683$	−6.13932e11	−0.107951	−0.0539756	+	0.998542i	$0.517189\pi$
$684$	0	0
$685$	−1.63526e12	−0.283779
$686$	0	0
$687$	−1.88730e13	−3.23248
$688$	0	0
$689$	−8.69576e11	−0.147001
$690$	0	0
$691$	−7.98229e12	−1.33191	−0.665957	−	0.745990i	$-0.731977\pi$
$691$	−7.98229e12	−1.33191	−0.665957	+	0.745990i	$0.731977\pi$
$692$	0	0
$693$	8.68527e12	1.43049
$694$	0	0
$695$	1.91711e12	0.311684
$696$	0	0
$697$	9.21660e10	0.0147919
$698$	0	0
$699$	−9.66017e12	−1.53052
$700$	0	0
$701$	5.04751e12	0.789490	0.394745	−	0.918791i	$-0.370833\pi$
$701$	5.04751e12	0.789490	0.394745	+	0.918791i	$0.370833\pi$
$702$	0	0
$703$	−1.73761e12	−0.268321
$704$	0	0
$705$	−1.23692e13	−1.88577
$706$	0	0
$707$	3.47627e12	0.523270
$708$	0	0
$709$	1.76603e12	0.262477	0.131238	−	0.991351i	$-0.458105\pi$
$709$	1.76603e12	0.262477	0.131238	+	0.991351i	$0.458105\pi$
$710$	0	0
$711$	9.80607e12	1.43907
$712$	0	0
$713$	7.73075e11	0.112026
$714$	0	0
$715$	3.30857e12	0.473437
$716$	0	0
$717$	6.01338e12	0.849733
$718$	0	0
$719$	−1.34595e12	−0.187823	−0.0939116	−	0.995581i	$-0.529937\pi$
$719$	−1.34595e12	−0.187823	−0.0939116	+	0.995581i	$0.529937\pi$
$720$	0	0
$721$	−6.83143e11	−0.0941462
$722$	0	0
$723$	5.98046e12	0.813977
$724$	0	0
$725$	4.06127e11	0.0545935
$726$	0	0
$727$	1.04449e13	1.38676	0.693379	−	0.720573i	$-0.256121\pi$
$727$	1.04449e13	1.38676	0.693379	+	0.720573i	$0.256121\pi$
$728$	0	0
$729$	−1.17483e13	−1.54065
$730$	0	0
$731$	−2.86491e11	−0.0371093
$732$	0	0
$733$	6.39102e12	0.817716	0.408858	−	0.912598i	$-0.365927\pi$
$733$	6.39102e12	0.817716	0.408858	+	0.912598i	$0.365927\pi$
$734$	0	0
$735$	−9.89239e12	−1.25028
$736$	0	0
$737$	−2.19957e13	−2.74621
$738$	0	0
$739$	−3.46900e12	−0.427863	−0.213931	−	0.976849i	$-0.568627\pi$
$739$	−3.46900e12	−0.427863	−0.213931	+	0.976849i	$0.568627\pi$
$740$	0	0
$741$	2.08581e12	0.254151
$742$	0	0
$743$	4.84775e12	0.583566	0.291783	−	0.956485i	$-0.405751\pi$
$743$	4.84775e12	0.583566	0.291783	+	0.956485i	$0.405751\pi$
$744$	0	0
$745$	−2.45871e12	−0.292418
$746$	0	0
$747$	2.29847e13	2.70082
$748$	0	0
$749$	2.90797e12	0.337615
$750$	0	0
$751$	−1.45382e13	−1.66775	−0.833874	−	0.551955i	$-0.813882\pi$
$751$	−1.45382e13	−1.66775	−0.833874	+	0.551955i	$0.813882\pi$
$752$	0	0
$753$	1.02382e13	1.16051
$754$	0	0
$755$	−1.43694e13	−1.60945
$756$	0	0
$757$	2.50225e12	0.276948	0.138474	−	0.990366i	$-0.455780\pi$
$757$	2.50225e12	0.276948	0.138474	+	0.990366i	$0.455780\pi$
$758$	0	0
$759$	1.27914e13	1.39904
$760$	0	0
$761$	1.25772e13	1.35941	0.679707	−	0.733484i	$-0.262107\pi$
$761$	1.25772e13	1.35941	0.679707	+	0.733484i	$0.262107\pi$
$762$	0	0
$763$	−7.82736e12	−0.836094
$764$	0	0
$765$	−1.66031e13	−1.75272
$766$	0	0
$767$	4.78323e12	0.499048
$768$	0	0
$769$	6.04403e12	0.623244	0.311622	−	0.950206i	$-0.399128\pi$
$769$	6.04403e12	0.623244	0.311622	+	0.950206i	$0.399128\pi$
$770$	0	0
$771$	2.96943e13	3.02641
$772$	0	0
$773$	6.97559e12	0.702705	0.351352	−	0.936243i	$-0.385722\pi$
$773$	6.97559e12	0.702705	0.351352	+	0.936243i	$0.385722\pi$
$774$	0	0
$775$	1.12843e11	0.0112361
$776$	0	0
$777$	−3.83944e12	−0.377897
$778$	0	0
$779$	8.11022e10	0.00789068
$780$	0	0
$781$	1.80927e13	1.74010
$782$	0	0
$783$	−1.42111e13	−1.35113
$784$	0	0
$785$	−1.39785e13	−1.31385
$786$	0	0
$787$	−7.96379e12	−0.740003	−0.370001	−	0.929031i	$-0.620643\pi$
$787$	−7.96379e12	−0.740003	−0.370001	+	0.929031i	$0.620643\pi$
$788$	0	0
$789$	−1.17144e13	−1.07615
$790$	0	0
$791$	−6.58665e12	−0.598233
$792$	0	0
$793$	−1.95752e12	−0.175783
$794$	0	0
$795$	9.61667e12	0.853833
$796$	0	0
$797$	1.07855e13	0.946845	0.473422	−	0.880836i	$-0.343018\pi$
$797$	1.07855e13	0.946845	0.473422	+	0.880836i	$0.343018\pi$
$798$	0	0
$799$	1.40267e13	1.21757
$800$	0	0
$801$	3.86461e13	3.31711
$802$	0	0
$803$	−2.71166e13	−2.30152
$804$	0	0
$805$	2.66185e12	0.223410
$806$	0	0
$807$	1.87295e13	1.55452
$808$	0	0
$809$	5.85843e12	0.480854	0.240427	−	0.970667i	$-0.422713\pi$
$809$	5.85843e12	0.480854	0.240427	+	0.970667i	$0.422713\pi$
$810$	0	0
$811$	−1.03329e12	−0.0838744	−0.0419372	−	0.999120i	$-0.513353\pi$
$811$	−1.03329e12	−0.0838744	−0.0419372	+	0.999120i	$0.513353\pi$
$812$	0	0
$813$	−7.48228e12	−0.600657
$814$	0	0
$815$	6.03551e11	0.0479187
$816$	0	0
$817$	−2.52100e11	−0.0197958
$818$	0	0
$819$	2.91910e12	0.226710
$820$	0	0
$821$	−5.24141e12	−0.402628	−0.201314	−	0.979527i	$-0.564521\pi$
$821$	−5.24141e12	−0.402628	−0.201314	+	0.979527i	$0.564521\pi$
$822$	0	0
$823$	1.63129e13	1.23946	0.619729	−	0.784816i	$-0.287243\pi$
$823$	1.63129e13	1.23946	0.619729	+	0.784816i	$0.287243\pi$
$824$	0	0
$825$	1.86712e12	0.140323
$826$	0	0
$827$	1.99959e13	1.48650	0.743252	−	0.669012i	$-0.233282\pi$
$827$	1.99959e13	1.48650	0.743252	+	0.669012i	$0.233282\pi$
$828$	0	0
$829$	−6.26388e12	−0.460626	−0.230313	−	0.973117i	$-0.573975\pi$
$829$	−6.26388e12	−0.460626	−0.230313	+	0.973117i	$0.573975\pi$
$830$	0	0
$831$	2.16860e13	1.57752
$832$	0	0
$833$	1.12180e13	0.807262
$834$	0	0
$835$	−3.06083e12	−0.217897
$836$	0	0
$837$	−3.94856e12	−0.278083
$838$	0	0
$839$	−1.52462e13	−1.06227	−0.531133	−	0.847289i	$-0.678233\pi$
$839$	−1.52462e13	−1.06227	−0.531133	+	0.847289i	$0.678233\pi$
$840$	0	0
$841$	3.83559e12	0.264393
$842$	0	0
$843$	2.77067e13	1.88956
$844$	0	0
$845$	1.11200e12	0.0750325
$846$	0	0
$847$	−1.46180e13	−0.975917
$848$	0	0
$849$	−4.23981e13	−2.80067
$850$	0	0
$851$	−3.58147e12	−0.234088
$852$	0	0
$853$	1.61356e13	1.04355	0.521775	−	0.853083i	$-0.325270\pi$
$853$	1.61356e13	1.04355	0.521775	+	0.853083i	$0.325270\pi$
$854$	0	0
$855$	−1.46100e13	−0.934982
$856$	0	0
$857$	2.75401e12	0.174402	0.0872011	−	0.996191i	$-0.472208\pi$
$857$	2.75401e12	0.174402	0.0872011	+	0.996191i	$0.472208\pi$
$858$	0	0
$859$	−6.38891e12	−0.400366	−0.200183	−	0.979758i	$-0.564154\pi$
$859$	−6.38891e12	−0.400366	−0.200183	+	0.979758i	$0.564154\pi$
$860$	0	0
$861$	1.79204e11	0.0111131
$862$	0	0
$863$	−9.56558e11	−0.0587033	−0.0293517	−	0.999569i	$-0.509344\pi$
$863$	−9.56558e11	−0.0587033	−0.0293517	+	0.999569i	$0.509344\pi$
$864$	0	0
$865$	5.20595e12	0.316175
$866$	0	0
$867$	2.24936e12	0.135199
$868$	0	0
$869$	−2.45063e13	−1.45777
$870$	0	0
$871$	−7.39268e12	−0.435231
$872$	0	0
$873$	−1.24399e13	−0.724856
$874$	0	0
$875$	8.39126e12	0.483940
$876$	0	0
$877$	7.09696e12	0.405111	0.202556	−	0.979271i	$-0.435075\pi$
$877$	7.09696e12	0.405111	0.202556	+	0.979271i	$0.435075\pi$
$878$	0	0
$879$	−3.63092e13	−2.05148
$880$	0	0
$881$	−2.00749e13	−1.12270	−0.561348	−	0.827580i	$-0.689717\pi$
$881$	−2.00749e13	−1.12270	−0.561348	+	0.827580i	$0.689717\pi$
$882$	0	0
$883$	7.53737e12	0.417250	0.208625	−	0.977996i	$-0.433101\pi$
$883$	7.53737e12	0.417250	0.208625	+	0.977996i	$0.433101\pi$
$884$	0	0
$885$	−5.28980e13	−2.89864
$886$	0	0
$887$	−3.17892e13	−1.72434	−0.862172	−	0.506616i	$-0.830896\pi$
$887$	−3.17892e13	−1.72434	−0.862172	+	0.506616i	$0.830896\pi$
$888$	0	0
$889$	−6.58221e12	−0.353439
$890$	0	0
$891$	−8.45803e12	−0.449593
$892$	0	0
$893$	1.23429e13	0.649510
$894$	0	0
$895$	−3.61958e13	−1.88562
$896$	0	0
$897$	4.29915e12	0.221726
$898$	0	0
$899$	5.09656e12	0.260231
$900$	0	0
$901$	−1.09054e13	−0.551288
$902$	0	0
$903$	−5.57041e11	−0.0278800
$904$	0	0
$905$	−1.67948e13	−0.832253
$906$	0	0
$907$	−1.54349e13	−0.757306	−0.378653	−	0.925539i	$-0.623613\pi$
$907$	−1.54349e13	−0.757306	−0.378653	+	0.925539i	$0.623613\pi$
$908$	0	0
$909$	−3.93268e13	−1.91052
$910$	0	0
$911$	−2.54205e13	−1.22279	−0.611395	−	0.791326i	$-0.709391\pi$
$911$	−2.54205e13	−1.22279	−0.611395	+	0.791326i	$0.709391\pi$
$912$	0	0
$913$	−5.74410e13	−2.73592
$914$	0	0
$915$	2.16482e13	1.02100
$916$	0	0
$917$	5.73847e12	0.268000
$918$	0	0
$919$	−1.15892e13	−0.535963	−0.267982	−	0.963424i	$-0.586357\pi$
$919$	−1.15892e13	−0.535963	−0.267982	+	0.963424i	$0.586357\pi$
$920$	0	0
$921$	3.94159e12	0.180511
$922$	0	0
$923$	6.08090e12	0.275778
$924$	0	0
$925$	−5.22775e11	−0.0234788
$926$	0	0
$927$	7.72835e12	0.343738
$928$	0	0
$929$	−3.69726e13	−1.62858	−0.814289	−	0.580459i	$-0.802873\pi$
$929$	−3.69726e13	−1.62858	−0.814289	+	0.580459i	$0.802873\pi$
$930$	0	0
$931$	9.87140e12	0.430631
$932$	0	0
$933$	5.11149e13	2.20841
$934$	0	0
$935$	4.14928e13	1.77550
$936$	0	0
$937$	−2.35152e13	−0.996601	−0.498301	−	0.867004i	$-0.666042\pi$
$937$	−2.35152e13	−0.996601	−0.498301	+	0.867004i	$0.666042\pi$
$938$	0	0
$939$	4.00472e13	1.68103
$940$	0	0
$941$	−2.49239e13	−1.03625	−0.518123	−	0.855306i	$-0.673369\pi$
$941$	−2.49239e13	−1.03625	−0.518123	+	0.855306i	$0.673369\pi$
$942$	0	0
$943$	1.67163e11	0.00688397
$944$	0	0
$945$	−1.35957e13	−0.554573
$946$	0	0
$947$	3.07652e13	1.24304	0.621519	−	0.783399i	$-0.286516\pi$
$947$	3.07652e13	1.24304	0.621519	+	0.783399i	$0.286516\pi$
$948$	0	0
$949$	−9.11381e12	−0.364756
$950$	0	0
$951$	−4.33846e13	−1.71998
$952$	0	0
$953$	−1.50639e13	−0.591586	−0.295793	−	0.955252i	$-0.595584\pi$
$953$	−1.50639e13	−0.591586	−0.295793	+	0.955252i	$0.595584\pi$
$954$	0	0
$955$	−4.63149e13	−1.80180
$956$	0	0
$957$	8.43283e13	3.24990
$958$	0	0
$959$	3.60562e12	0.137656
$960$	0	0
$961$	−2.50235e13	−0.946441
$962$	0	0
$963$	−3.28976e13	−1.23267
$964$	0	0
$965$	1.94056e13	0.720367
$966$	0	0
$967$	5.17914e11	0.0190475	0.00952377	−	0.999955i	$-0.496968\pi$
$967$	5.17914e11	0.0190475	0.00952377	+	0.999955i	$0.496968\pi$
$968$	0	0
$969$	2.61582e13	0.953126
$970$	0	0
$971$	1.97223e13	0.711987	0.355993	−	0.934488i	$-0.384143\pi$
$971$	1.97223e13	0.711987	0.355993	+	0.934488i	$0.384143\pi$
$972$	0	0
$973$	−4.22706e12	−0.151193
$974$	0	0
$975$	6.27532e11	0.0222390
$976$	0	0
$977$	−3.01410e13	−1.05836	−0.529178	−	0.848511i	$-0.677500\pi$
$977$	−3.01410e13	−1.05836	−0.529178	+	0.848511i	$0.677500\pi$
$978$	0	0
$979$	−9.65805e13	−3.36022
$980$	0	0
$981$	8.85504e13	3.05267
$982$	0	0
$983$	3.44393e13	1.17642	0.588211	−	0.808708i	$-0.299833\pi$
$983$	3.44393e13	1.17642	0.588211	+	0.808708i	$0.299833\pi$
$984$	0	0
$985$	4.37844e13	1.48203
$986$	0	0
$987$	2.72730e13	0.914756
$988$	0	0
$989$	−5.19614e11	−0.0172702
$990$	0	0
$991$	−2.44194e13	−0.804273	−0.402137	−	0.915580i	$-0.631732\pi$
$991$	−2.44194e13	−0.804273	−0.402137	+	0.915580i	$0.631732\pi$
$992$	0	0
$993$	−1.67959e13	−0.548192
$994$	0	0
$995$	−4.50516e13	−1.45716
$996$	0	0
$997$	−1.69672e13	−0.543853	−0.271926	−	0.962318i	$-0.587661\pi$
$997$	−1.69672e13	−0.543853	−0.271926	+	0.962318i	$0.587661\pi$
$998$	0	0
$999$	1.82927e13	0.581078

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	208.10.a.l.1.7		7
4.3	odd	2		104.10.a.c.1.1	✓	7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
104.10.a.c.1.1	✓	7	4.3	odd	2
208.10.a.l.1.7		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)

\(f(q)\)	\(=\)	\(q+231.704 q^{3} +1363.19 q^{5} -3005.73 q^{7} +34003.6 q^{9} -84978.4 q^{11} -28561.0 q^{13} +315857. q^{15} -358184. q^{17} -315187. q^{19} -696439. q^{21} -649646. q^{23} -94826.5 q^{25} +3.31814e6 q^{27} -4.28284e6 q^{29} -1.18999e6 q^{31} -1.96898e7 q^{33} -4.09739e6 q^{35} +5.51296e6 q^{37} -6.61769e6 q^{39} -257315. q^{41} +799842. q^{43} +4.63535e7 q^{45} -3.91606e7 q^{47} -3.13192e7 q^{49} -8.29926e7 q^{51} +3.04463e7 q^{53} -1.15842e8 q^{55} -7.30300e7 q^{57} -1.67474e8 q^{59} +6.85381e7 q^{61} -1.02206e8 q^{63} -3.89342e7 q^{65} +2.58838e8 q^{67} -1.50525e8 q^{69} -2.12909e8 q^{71} +3.19100e8 q^{73} -2.19717e7 q^{75} +2.55422e8 q^{77} +2.88383e8 q^{79} +9.95316e7 q^{81} +6.75948e8 q^{83} -4.88274e8 q^{85} -9.92350e8 q^{87} +1.13653e9 q^{89} +8.58467e7 q^{91} -2.75726e8 q^{93} -4.29661e8 q^{95} -3.65840e8 q^{97} -2.88957e9 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q - 13 q^{3} - 801 q^{5} + 1091 q^{7} + 87864 q^{9} - 32218 q^{11} - 199927 q^{13} - 58323 q^{15} - 870531 q^{17} + 128950 q^{19} - 719663 q^{21} - 2198844 q^{23} + 992010 q^{25} - 5441383 q^{27} + 6327710 q^{29}+ \cdots - 5204667600 q^{99}+O(q^{100}) \) 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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