LMFDB - Embedded newform 338.8.a.m.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [338,8,Mod(1,338)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("338.1"); S:= CuspForms(chi, 8); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(338, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 8, names="a")

Level:	$ N $	$=$	$ 338 = 2 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	338.a (trivial)

Newform invariants

comment:select newform

sage:traces = [8,-64,0] 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:	$105.586138614$
Analytic rank:	$1$
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} - 11090 x^{6} - 22320 x^{5} + 30742777 x^{4} + 246003120 x^{3} - 12575752488 x^{2} + \cdots + 137733023376 $ x^8 - 11090x^6 - 22320x^5 + 30742777x^4 + 246003120x^3 - 12575752488x^2 + 49433716800x + 137733023376
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{6}\cdot 13^{3} $
Twist minimal:	no (minimal twist has level 26)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-32.3527$ of defining polynomial
Character	$\chi$	$=$	338.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-8.00000 q^{2} -32.3527 q^{3} +64.0000 q^{4} +211.408 q^{5} +258.821 q^{6} +734.154 q^{7} -512.000 q^{8} -1140.31 q^{9} -1691.27 q^{10} +8597.18 q^{11} -2070.57 q^{12} -5873.23 q^{14} -6839.62 q^{15} +4096.00 q^{16} -16338.0 q^{17} +9122.45 q^{18} -3502.39 q^{19} +13530.1 q^{20} -23751.8 q^{21} -68777.4 q^{22} -65276.4 q^{23} +16564.6 q^{24} -33431.5 q^{25} +107647. q^{27} +46985.8 q^{28} -112591. q^{29} +54716.9 q^{30} +176071. q^{31} -32768.0 q^{32} -278142. q^{33} +130704. q^{34} +155206. q^{35} -72979.6 q^{36} -499275. q^{37} +28019.1 q^{38} -108241. q^{40} -289155. q^{41} +190015. q^{42} -402020. q^{43} +550219. q^{44} -241070. q^{45} +522211. q^{46} -64966.5 q^{47} -132516. q^{48} -284562. q^{49} +267452. q^{50} +528576. q^{51} +263903. q^{53} -861177. q^{54} +1.81751e6 q^{55} -375887. q^{56} +113312. q^{57} +900731. q^{58} +2.57104e6 q^{59} -437736. q^{60} -538951. q^{61} -1.40857e6 q^{62} -837160. q^{63} +262144. q^{64} +2.22513e6 q^{66} +891050. q^{67} -1.04563e6 q^{68} +2.11187e6 q^{69} -1.24165e6 q^{70} -3.95520e6 q^{71} +583837. q^{72} +2.77895e6 q^{73} +3.99420e6 q^{74} +1.08160e6 q^{75} -224153. q^{76} +6.31165e6 q^{77} +4.53826e6 q^{79} +865928. q^{80} -988822. q^{81} +2.31324e6 q^{82} +2.61022e6 q^{83} -1.52012e6 q^{84} -3.45398e6 q^{85} +3.21616e6 q^{86} +3.64263e6 q^{87} -4.40176e6 q^{88} +7.87803e6 q^{89} +1.92856e6 q^{90} -4.17769e6 q^{92} -5.69636e6 q^{93} +519732. q^{94} -740435. q^{95} +1.06013e6 q^{96} -9.14006e6 q^{97} +2.27649e6 q^{98} -9.80341e6 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 64 q^{2} + 512 q^{4} - 304 q^{5} - 2160 q^{7} - 4096 q^{8} + 4684 q^{9} + 2432 q^{10} - 2160 q^{11} + 17280 q^{14} - 21296 q^{15} + 32768 q^{16} + 41520 q^{17} - 37472 q^{18} - 66240 q^{19} - 19456 q^{20}+ \cdots - 44172000 q^{99}+O(q^{100}) $ 8 * q - 64 * q^2 + 512 * q^4 - 304 * q^5 - 2160 * q^7 - 4096 * q^8 + 4684 * q^9 + 2432 * q^10 - 2160 * q^11 + 17280 * q^14 - 21296 * q^15 + 32768 * q^16 + 41520 * q^17 - 37472 * q^18 - 66240 * q^19 - 19456 * q^20 + 51040 * q^21 + 17280 * q^22 - 7560 * q^23 + 136980 * q^25 + 66960 * q^27 - 138240 * q^28 + 346056 * q^29 + 170368 * q^30 + 346320 * q^31 - 262144 * q^32 - 496320 * q^33 - 332160 * q^34 + 283248 * q^35 + 299776 * q^36 - 142800 * q^37 + 529920 * q^38 + 155648 * q^40 - 1525280 * q^41 - 408320 * q^42 + 285360 * q^43 - 138240 * q^44 + 254416 * q^45 + 60480 * q^46 - 1787952 * q^47 - 2131660 * q^49 - 1095840 * q^50 - 478640 * q^51 + 1833300 * q^53 - 535680 * q^54 + 1091472 * q^55 + 1105920 * q^56 + 1694208 * q^57 - 2768448 * q^58 + 1034000 * q^59 - 1362944 * q^60 + 4626000 * q^61 - 2770560 * q^62 - 3577760 * q^63 + 2097152 * q^64 + 3970560 * q^66 - 4959456 * q^67 + 2657280 * q^68 + 18027316 * q^69 - 2265984 * q^70 - 2591040 * q^71 - 2398208 * q^72 - 5390592 * q^73 + 1142400 * q^74 - 10898032 * q^75 - 4239360 * q^76 + 10288740 * q^77 + 17071744 * q^79 - 1245184 * q^80 + 15745864 * q^81 + 12202240 * q^82 - 17216416 * q^83 + 3266560 * q^84 + 19975440 * q^85 - 2282880 * q^86 + 10460440 * q^87 + 1105920 * q^88 + 12984960 * q^89 - 2035328 * q^90 - 483840 * q^92 + 8072768 * q^93 + 14303616 * q^94 + 33837504 * q^95 - 13500864 * q^97 + 17053280 * q^98 - 44172000 * 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 $

$2$	−8.00000	−0.707107
$2$	−8.00000	−0.707107
$3$	−32.3527	−0.691808	−0.345904	−	0.938270i	$-0.612428\pi$
$3$	−32.3527	−0.691808	−0.345904	+	0.938270i	$0.612428\pi$
$4$	64.0000	0.500000
$5$	211.408	0.756357	0.378179	−	0.925733i	$-0.376551\pi$
$5$	211.408	0.756357	0.378179	+	0.925733i	$0.376551\pi$
$6$	258.821	0.489182
$7$	734.154	0.808991	0.404496	−	0.914540i	$-0.367447\pi$
$7$	734.154	0.808991	0.404496	+	0.914540i	$0.367447\pi$
$8$	−512.000	−0.353553
$9$	−1140.31	−0.521402
$10$	−1691.27	−0.534825
$11$	8597.18	1.94752	0.973759	−	0.227581i	$-0.0730816\pi$
$11$	8597.18	1.94752	0.973759	+	0.227581i	$0.0730816\pi$
$12$	−2070.57	−0.345904
$13$	0	0
$13$	0	0
$14$	−5873.23	−0.572043
$15$	−6839.62	−0.523254
$16$	4096.00	0.250000
$17$	−16338.0	−0.806541	−0.403271	−	0.915081i	$-0.632127\pi$
$17$	−16338.0	−0.806541	−0.403271	+	0.915081i	$0.632127\pi$
$18$	9122.45	0.368687
$19$	−3502.39	−0.117146	−0.0585730	−	0.998283i	$-0.518655\pi$
$19$	−3502.39	−0.117146	−0.0585730	+	0.998283i	$0.518655\pi$
$20$	13530.1	0.378179
$21$	−23751.8	−0.559666
$22$	−68777.4	−1.37710
$23$	−65276.4	−1.11869	−0.559344	−	0.828936i	$-0.688947\pi$
$23$	−65276.4	−1.11869	−0.559344	+	0.828936i	$0.688947\pi$
$24$	16564.6	0.244591
$25$	−33431.5	−0.427924
$26$	0	0
$27$	107647.	1.05252
$28$	46985.8	0.404496
$29$	−112591.	−0.857259	−0.428630	−	0.903480i	$-0.641003\pi$
$29$	−112591.	−0.857259	−0.428630	+	0.903480i	$0.641003\pi$
$30$	54716.9	0.369996
$31$	176071.	1.06150	0.530752	−	0.847527i	$-0.321909\pi$
$31$	176071.	1.06150	0.530752	+	0.847527i	$0.321909\pi$
$32$	−32768.0	−0.176777
$33$	−278142.	−1.34731
$34$	130704.	0.570311
$35$	155206.	0.611886
$36$	−72979.6	−0.260701
$37$	−499275.	−1.62044	−0.810222	−	0.586123i	$-0.800654\pi$
$37$	−499275.	−1.62044	−0.810222	+	0.586123i	$0.800654\pi$
$38$	28019.1	0.0828347
$39$	0	0
$40$	−108241.	−0.267413
$41$	−289155.	−0.655219	−0.327610	−	0.944813i	$-0.606243\pi$
$41$	−289155.	−0.655219	−0.327610	+	0.944813i	$0.606243\pi$
$42$	190015.	0.395744
$43$	−402020.	−0.771096	−0.385548	−	0.922688i	$-0.625987\pi$
$43$	−402020.	−0.771096	−0.385548	+	0.922688i	$0.625987\pi$
$44$	550219.	0.973759
$45$	−241070.	−0.394366
$46$	522211.	0.791032
$47$	−64966.5	−0.0912739	−0.0456370	−	0.998958i	$-0.514532\pi$
$47$	−64966.5	−0.0912739	−0.0456370	+	0.998958i	$0.514532\pi$
$48$	−132516.	−0.172952
$49$	−284562.	−0.345533
$50$	267452.	0.302588
$51$	528576.	0.557971
$52$	0	0
$53$	263903.	0.243489	0.121744	−	0.992561i	$-0.461151\pi$
$53$	263903.	0.243489	0.121744	+	0.992561i	$0.461151\pi$
$54$	−861177.	−0.744242
$55$	1.81751e6	1.47302
$56$	−375887.	−0.286022
$57$	113312.	0.0810425
$58$	900731.	0.606174
$59$	2.57104e6	1.62977	0.814887	−	0.579620i	$-0.196799\pi$
$59$	2.57104e6	1.62977	0.814887	+	0.579620i	$0.196799\pi$
$60$	−437736.	−0.261627
$61$	−538951.	−0.304015	−0.152007	−	0.988379i	$-0.548574\pi$
$61$	−538951.	−0.304015	−0.152007	+	0.988379i	$0.548574\pi$
$62$	−1.40857e6	−0.750597
$63$	−837160.	−0.421810
$64$	262144.	0.125000
$65$	0	0
$66$	2.22513e6	0.952691
$67$	891050.	0.361943	0.180972	−	0.983488i	$-0.442076\pi$
$67$	891050.	0.361943	0.180972	+	0.983488i	$0.442076\pi$
$68$	−1.04563e6	−0.403271
$69$	2.11187e6	0.773917
$70$	−1.24165e6	−0.432669
$71$	−3.95520e6	−1.31149	−0.655744	−	0.754984i	$-0.727645\pi$
$71$	−3.95520e6	−1.31149	−0.655744	+	0.754984i	$0.727645\pi$
$72$	583837.	0.184343
$73$	2.77895e6	0.836087	0.418043	−	0.908427i	$-0.362716\pi$
$73$	2.77895e6	0.836087	0.418043	+	0.908427i	$0.362716\pi$
$74$	3.99420e6	1.14583
$75$	1.08160e6	0.296041
$76$	−224153.	−0.0585730
$77$	6.31165e6	1.57553
$78$	0	0
$79$	4.53826e6	1.03561	0.517803	−	0.855500i	$-0.326750\pi$
$79$	4.53826e6	1.03561	0.517803	+	0.855500i	$0.326750\pi$
$80$	865928.	0.189089
$81$	−988822.	−0.206738
$82$	2.31324e6	0.463310
$83$	2.61022e6	0.501076	0.250538	−	0.968107i	$-0.419393\pi$
$83$	2.61022e6	0.501076	0.250538	+	0.968107i	$0.419393\pi$
$84$	−1.52012e6	−0.279833
$85$	−3.45398e6	−0.610033
$86$	3.21616e6	0.545247
$87$	3.64263e6	0.593059
$88$	−4.40176e6	−0.688552
$89$	7.87803e6	1.18455	0.592274	−	0.805737i	$-0.298230\pi$
$89$	7.87803e6	1.18455	0.592274	+	0.805737i	$0.298230\pi$
$90$	1.92856e6	0.278859
$91$	0	0
$92$	−4.17769e6	−0.559344
$93$	−5.69636e6	−0.734357
$94$	519732.	0.0645404
$95$	−740435.	−0.0886042
$96$	1.06013e6	0.122296
$97$	−9.14006e6	−1.01683	−0.508415	−	0.861112i	$-0.669768\pi$
$97$	−9.14006e6	−1.01683	−0.508415	+	0.861112i	$0.669768\pi$
$98$	2.27649e6	0.244329
$99$	−9.80341e6	−1.01544
$100$	−2.13962e6	−0.213962
$101$	−5.74830e6	−0.555155	−0.277578	−	0.960703i	$-0.589532\pi$
$101$	−5.74830e6	−0.555155	−0.277578	+	0.960703i	$0.589532\pi$
$102$	−4.22861e6	−0.394545
$103$	9.21760e6	0.831166	0.415583	−	0.909555i	$-0.363578\pi$
$103$	9.21760e6	0.831166	0.415583	+	0.909555i	$0.363578\pi$
$104$	0	0
$105$	−5.02133e6	−0.423308
$106$	−2.11122e6	−0.172173
$107$	−1.67150e7	−1.31905	−0.659526	−	0.751682i	$-0.729243\pi$
$107$	−1.67150e7	−1.31905	−0.659526	+	0.751682i	$0.729243\pi$
$108$	6.88942e6	0.526259
$109$	−1.42886e7	−1.05681	−0.528403	−	0.848993i	$-0.677209\pi$
$109$	−1.42886e7	−1.05681	−0.528403	+	0.848993i	$0.677209\pi$
$110$	−1.45401e7	−1.04158
$111$	1.61529e7	1.12104
$112$	3.00709e6	0.202248
$113$	−2.19653e7	−1.43206	−0.716032	−	0.698067i	$-0.754044\pi$
$113$	−2.19653e7	−1.43206	−0.716032	+	0.698067i	$0.754044\pi$
$114$	−906493.	−0.0573057
$115$	−1.38000e7	−0.846128
$116$	−7.20585e6	−0.428630
$117$	0	0
$118$	−2.05683e7	−1.15242
$119$	−1.19946e7	−0.652485
$120$	3.50188e6	0.184998
$121$	5.44243e7	2.79283
$122$	4.31161e6	0.214971
$123$	9.35492e6	0.453286
$124$	1.12685e7	0.530752
$125$	−2.35840e7	−1.08002
$126$	6.69728e6	0.298264
$127$	−2.14011e7	−0.927093	−0.463546	−	0.886073i	$-0.653423\pi$
$127$	−2.14011e7	−0.927093	−0.463546	+	0.886073i	$0.653423\pi$
$128$	−2.09715e6	−0.0883883
$129$	1.30064e7	0.533450
$130$	0	0
$131$	2.79550e7	1.08645	0.543226	−	0.839587i	$-0.317203\pi$
$131$	2.79550e7	1.08645	0.543226	+	0.839587i	$0.317203\pi$
$132$	−1.78011e7	−0.673654
$133$	−2.57129e6	−0.0947700
$134$	−7.12840e6	−0.255932
$135$	2.27575e7	0.796079
$136$	8.36503e6	0.285155
$137$	−4.47783e7	−1.48780	−0.743902	−	0.668289i	$-0.767027\pi$
$137$	−4.47783e7	−1.48780	−0.743902	+	0.668289i	$0.767027\pi$
$138$	−1.68949e7	−0.547242
$139$	2.39446e7	0.756235	0.378117	−	0.925758i	$-0.376572\pi$
$139$	2.39446e7	0.756235	0.378117	+	0.925758i	$0.376572\pi$
$140$	9.93319e6	0.305943
$141$	2.10184e6	0.0631440
$142$	3.16416e7	0.927362
$143$	0	0
$144$	−4.67069e6	−0.130350
$145$	−2.38028e7	−0.648394
$146$	−2.22316e7	−0.591203
$147$	9.20632e6	0.239043
$148$	−3.19536e7	−0.810222
$149$	−3.93169e7	−0.973705	−0.486852	−	0.873484i	$-0.661855\pi$
$149$	−3.93169e7	−0.973705	−0.486852	+	0.873484i	$0.661855\pi$
$150$	−8.65279e6	−0.209333
$151$	−5.45163e7	−1.28857	−0.644284	−	0.764786i	$-0.722845\pi$
$151$	−5.45163e7	−1.28857	−0.644284	+	0.764786i	$0.722845\pi$
$152$	1.79322e6	0.0414173
$153$	1.86303e7	0.420532
$154$	−5.04932e7	−1.11406
$155$	3.72229e7	0.802877
$156$	0	0
$157$	−6.58173e7	−1.35735	−0.678674	−	0.734440i	$-0.737445\pi$
$157$	−6.58173e7	−1.35735	−0.678674	+	0.734440i	$0.737445\pi$
$158$	−3.63061e7	−0.732285
$159$	−8.53796e6	−0.168447
$160$	−6.92743e6	−0.133706
$161$	−4.79229e7	−0.905009
$162$	7.91058e6	0.146186
$163$	6.52159e7	1.17950	0.589748	−	0.807587i	$-0.299227\pi$
$163$	6.52159e7	1.17950	0.589748	+	0.807587i	$0.299227\pi$
$164$	−1.85059e7	−0.327610
$165$	−5.88014e7	−1.01905
$166$	−2.08817e7	−0.354314
$167$	6.32815e7	1.05140	0.525701	−	0.850669i	$-0.323803\pi$
$167$	6.32815e7	1.05140	0.525701	+	0.850669i	$0.323803\pi$
$168$	1.21609e7	0.197872
$169$	0	0
$170$	2.76318e7	0.431359
$171$	3.99380e6	0.0610801
$172$	−2.57293e7	−0.385548
$173$	−7.69158e7	−1.12942	−0.564709	−	0.825290i	$-0.691011\pi$
$173$	−7.69158e7	−1.12942	−0.564709	+	0.825290i	$0.691011\pi$
$174$	−2.91410e7	−0.419356
$175$	−2.45439e7	−0.346187
$176$	3.52140e7	0.486880
$177$	−8.31801e7	−1.12749
$178$	−6.30242e7	−0.837602
$179$	−8.64732e7	−1.12693	−0.563464	−	0.826141i	$-0.690532\pi$
$179$	−8.64732e7	−1.12693	−0.563464	+	0.826141i	$0.690532\pi$
$180$	−1.54285e7	−0.197183
$181$	8.42468e7	1.05603	0.528017	−	0.849234i	$-0.322936\pi$
$181$	8.42468e7	1.05603	0.528017	+	0.849234i	$0.322936\pi$
$182$	0	0
$183$	1.74365e7	0.210320
$184$	3.34215e7	0.395516
$185$	−1.05551e8	−1.22563
$186$	4.55709e7	0.519269
$187$	−1.40460e8	−1.57075
$188$	−4.15785e6	−0.0456370
$189$	7.90296e7	0.851478
$190$	5.92348e6	0.0626526
$191$	−3.21926e7	−0.334302	−0.167151	−	0.985931i	$-0.553457\pi$
$191$	−3.21926e7	−0.334302	−0.167151	+	0.985931i	$0.553457\pi$
$192$	−8.48105e6	−0.0864760
$193$	−8.86497e7	−0.887620	−0.443810	−	0.896121i	$-0.646373\pi$
$193$	−8.86497e7	−0.887620	−0.443810	+	0.896121i	$0.646373\pi$
$194$	7.31205e7	0.719007
$195$	0	0
$196$	−1.82119e7	−0.172767
$197$	8.88151e6	0.0827666	0.0413833	−	0.999143i	$-0.486824\pi$
$197$	8.88151e6	0.0827666	0.0413833	+	0.999143i	$0.486824\pi$
$198$	7.84273e7	0.718024
$199$	−7.90457e7	−0.711037	−0.355519	−	0.934669i	$-0.615696\pi$
$199$	−7.90457e7	−0.711037	−0.355519	+	0.934669i	$0.615696\pi$
$200$	1.71170e7	0.151294
$201$	−2.88278e7	−0.250395
$202$	4.59864e7	0.392554
$203$	−8.26594e7	−0.693515
$204$	3.38289e7	0.278986
$205$	−6.11297e7	−0.495580
$206$	−7.37408e7	−0.587723
$207$	7.44351e7	0.583286
$208$	0	0
$209$	−3.01107e7	−0.228144
$210$	4.01706e7	0.299324
$211$	−2.17540e8	−1.59423	−0.797114	−	0.603828i	$-0.793641\pi$
$211$	−2.17540e8	−1.59423	−0.797114	+	0.603828i	$0.793641\pi$
$212$	1.68898e7	0.121744
$213$	1.27961e8	0.907297
$214$	1.33720e8	0.932711
$215$	−8.49904e7	−0.583224
$216$	−5.51154e7	−0.372121
$217$	1.29263e8	0.858748
$218$	1.14308e8	0.747275
$219$	−8.99065e7	−0.578411
$220$	1.16321e8	0.736510
$221$	0	0
$222$	−1.29223e8	−0.792692
$223$	−1.25862e8	−0.760022	−0.380011	−	0.924982i	$-0.624080\pi$
$223$	−1.25862e8	−0.760022	−0.380011	+	0.924982i	$0.624080\pi$
$224$	−2.40567e7	−0.143011
$225$	3.81222e7	0.223120
$226$	1.75722e8	1.01262
$227$	2.64141e8	1.49880	0.749402	−	0.662115i	$-0.230341\pi$
$227$	2.64141e8	1.49880	0.749402	+	0.662115i	$0.230341\pi$
$228$	7.25195e6	0.0405212
$229$	6.38112e7	0.351134	0.175567	−	0.984467i	$-0.443824\pi$
$229$	6.38112e7	0.351134	0.175567	+	0.984467i	$0.443824\pi$
$230$	1.10400e8	0.598303
$231$	−2.04199e8	−1.08996
$232$	5.76468e7	0.303087
$233$	1.58319e8	0.819947	0.409974	−	0.912097i	$-0.365538\pi$
$233$	1.58319e8	0.819947	0.409974	+	0.912097i	$0.365538\pi$
$234$	0	0
$235$	−1.37344e7	−0.0690357
$236$	1.64547e8	0.814887
$237$	−1.46825e8	−0.716441
$238$	9.59565e7	0.461376
$239$	−2.79618e8	−1.32487	−0.662434	−	0.749120i	$-0.730477\pi$
$239$	−2.79618e8	−1.32487	−0.662434	+	0.749120i	$0.730477\pi$
$240$	−2.80151e7	−0.130813
$241$	2.26300e8	1.04142	0.520710	−	0.853734i	$-0.325667\pi$
$241$	2.26300e8	1.04142	0.520710	+	0.853734i	$0.325667\pi$
$242$	−4.35395e8	−1.97483
$243$	−2.03433e8	−0.909495
$244$	−3.44929e7	−0.152007
$245$	−6.01587e7	−0.261347
$246$	−7.48393e7	−0.320521
$247$	0	0
$248$	−9.01484e7	−0.375299
$249$	−8.44474e7	−0.346648
$250$	1.88672e8	0.763690
$251$	2.16438e8	0.863925	0.431962	−	0.901892i	$-0.357821\pi$
$251$	2.16438e8	0.863925	0.431962	+	0.901892i	$0.357821\pi$
$252$	−5.35782e7	−0.210905
$253$	−5.61193e8	−2.17867
$254$	1.71209e8	0.655554
$255$	1.11745e8	0.422026
$256$	1.67772e7	0.0625000
$257$	2.43576e8	0.895094	0.447547	−	0.894260i	$-0.352298\pi$
$257$	2.43576e8	0.895094	0.447547	+	0.894260i	$0.352298\pi$
$258$	−1.04051e8	−0.377206
$259$	−3.66545e8	−1.31093
$260$	0	0
$261$	1.28389e8	0.446977
$262$	−2.23640e8	−0.768238
$263$	−5.04503e6	−0.0171009	−0.00855045	−	0.999963i	$-0.502722\pi$
$263$	−5.04503e6	−0.0171009	−0.00855045	+	0.999963i	$0.502722\pi$
$264$	1.42408e8	0.476345
$265$	5.57913e7	0.184164
$266$	2.05703e7	0.0670125
$267$	−2.54875e8	−0.819479
$268$	5.70272e7	0.180972
$269$	−4.65369e8	−1.45769	−0.728844	−	0.684680i	$-0.759942\pi$
$269$	−4.65369e8	−1.45769	−0.728844	+	0.684680i	$0.759942\pi$
$270$	−1.82060e8	−0.562913
$271$	3.31774e8	1.01263	0.506314	−	0.862349i	$-0.331008\pi$
$271$	3.31774e8	1.01263	0.506314	+	0.862349i	$0.331008\pi$
$272$	−6.69203e7	−0.201635
$273$	0	0
$274$	3.58226e8	1.05204
$275$	−2.87417e8	−0.833389
$276$	1.35159e8	0.386959
$277$	−4.08534e8	−1.15491	−0.577456	−	0.816422i	$-0.695955\pi$
$277$	−4.08534e8	−1.15491	−0.577456	+	0.816422i	$0.695955\pi$
$278$	−1.91557e8	−0.534739
$279$	−2.00775e8	−0.553471
$280$	−7.94655e7	−0.216334
$281$	6.26412e8	1.68418	0.842089	−	0.539339i	$-0.181326\pi$
$281$	6.26412e8	1.68418	0.842089	+	0.539339i	$0.181326\pi$
$282$	−1.68147e7	−0.0446496
$283$	−2.74870e7	−0.0720901	−0.0360450	−	0.999350i	$-0.511476\pi$
$283$	−2.74870e7	−0.0720901	−0.0360450	+	0.999350i	$0.511476\pi$
$284$	−2.53133e8	−0.655744
$285$	2.39550e7	0.0612971
$286$	0	0
$287$	−2.12284e8	−0.530066
$288$	3.73655e7	0.0921717
$289$	−1.43410e8	−0.349492
$290$	1.90422e8	0.458484
$291$	2.95705e8	0.703451
$292$	1.77853e8	0.418043
$293$	−1.85059e8	−0.429807	−0.214904	−	0.976635i	$-0.568944\pi$
$293$	−1.85059e8	−0.429807	−0.214904	+	0.976635i	$0.568944\pi$
$294$	−7.36506e7	−0.169029
$295$	5.43540e8	1.23269
$296$	2.55629e8	0.572914
$297$	9.25462e8	2.04980
$298$	3.14535e8	0.688513
$299$	0	0
$300$	6.92223e7	0.148021
$301$	−2.95144e8	−0.623809
$302$	4.36131e8	0.911155
$303$	1.85973e8	0.384061
$304$	−1.43458e7	−0.0292865
$305$	−1.13939e8	−0.229944
$306$	−1.49042e8	−0.297361
$307$	−2.08464e7	−0.0411194	−0.0205597	−	0.999789i	$-0.506545\pi$
$307$	−2.08464e7	−0.0411194	−0.0205597	+	0.999789i	$0.506545\pi$
$308$	4.03946e8	0.787763
$309$	−2.98214e8	−0.575007
$310$	−2.97783e8	−0.567720
$311$	−7.13088e7	−0.134426	−0.0672128	−	0.997739i	$-0.521411\pi$
$311$	−7.13088e7	−0.134426	−0.0672128	+	0.997739i	$0.521411\pi$
$312$	0	0
$313$	3.19037e8	0.588080	0.294040	−	0.955793i	$-0.405000\pi$
$313$	3.19037e8	0.588080	0.294040	+	0.955793i	$0.405000\pi$
$314$	5.26538e8	0.959790
$315$	−1.76982e8	−0.319039
$316$	2.90449e8	0.517803
$317$	3.53993e8	0.624147	0.312074	−	0.950058i	$-0.398976\pi$
$317$	3.53993e8	0.624147	0.312074	+	0.950058i	$0.398976\pi$
$318$	6.83037e7	0.119110
$319$	−9.67968e8	−1.66953
$320$	5.54194e7	0.0945447
$321$	5.40773e8	0.912531
$322$	3.83383e8	0.639938
$323$	5.72219e7	0.0944830
$324$	−6.32846e7	−0.103369
$325$	0	0
$326$	−5.21727e8	−0.834030
$327$	4.62273e8	0.731107
$328$	1.48047e8	0.231655
$329$	−4.76954e7	−0.0738398
$330$	4.70411e8	0.720575
$331$	−1.00884e9	−1.52907	−0.764533	−	0.644584i	$-0.777030\pi$
$331$	−1.00884e9	−1.52907	−0.764533	+	0.644584i	$0.777030\pi$
$332$	1.67054e8	0.250538
$333$	5.69327e8	0.844903
$334$	−5.06252e8	−0.743454
$335$	1.88375e8	0.273758
$336$	−9.72874e7	−0.139917
$337$	−6.85528e8	−0.975709	−0.487855	−	0.872925i	$-0.662220\pi$
$337$	−6.85528e8	−0.975709	−0.487855	+	0.872925i	$0.662220\pi$
$338$	0	0
$339$	7.10636e8	0.990714
$340$	−2.21055e8	−0.305017
$341$	1.51371e9	2.06730
$342$	−3.19504e7	−0.0431902
$343$	−8.13519e8	−1.08852
$344$	2.05834e8	0.272623
$345$	4.46466e8	0.585358
$346$	6.15327e8	0.798619
$347$	−4.98296e8	−0.640227	−0.320114	−	0.947379i	$-0.603721\pi$
$347$	−4.98296e8	−0.640227	−0.320114	+	0.947379i	$0.603721\pi$
$348$	2.33128e8	0.296529
$349$	2.80984e8	0.353828	0.176914	−	0.984226i	$-0.443388\pi$
$349$	2.80984e8	0.353828	0.176914	+	0.984226i	$0.443388\pi$
$350$	1.96351e8	0.244791
$351$	0	0
$352$	−2.81712e8	−0.344276
$353$	6.59832e8	0.798403	0.399201	−	0.916863i	$-0.369287\pi$
$353$	6.59832e8	0.798403	0.399201	+	0.916863i	$0.369287\pi$
$354$	6.65441e8	0.797256
$355$	−8.36162e8	−0.991953
$356$	5.04194e8	0.592274
$357$	3.88056e8	0.451394
$358$	6.91786e8	0.796858
$359$	−3.72973e7	−0.0425448	−0.0212724	−	0.999774i	$-0.506772\pi$
$359$	−3.72973e7	−0.0425448	−0.0212724	+	0.999774i	$0.506772\pi$
$360$	1.23428e8	0.139429
$361$	−8.81605e8	−0.986277
$362$	−6.73974e8	−0.746729
$363$	−1.76077e9	−1.93210
$364$	0	0
$365$	5.87494e8	0.632380
$366$	−1.39492e8	−0.148719
$367$	−7.20486e8	−0.760841	−0.380421	−	0.924814i	$-0.624221\pi$
$367$	−7.20486e8	−0.760841	−0.380421	+	0.924814i	$0.624221\pi$
$368$	−2.67372e8	−0.279672
$369$	3.29725e8	0.341632
$370$	8.44407e8	0.866655
$371$	1.93745e8	0.196980
$372$	−3.64567e8	−0.367179
$373$	−2.91004e8	−0.290348	−0.145174	−	0.989406i	$-0.546374\pi$
$373$	−2.91004e8	−0.290348	−0.145174	+	0.989406i	$0.546374\pi$
$374$	1.12368e9	1.11069
$375$	7.63004e8	0.747167
$376$	3.32628e7	0.0322702
$377$	0	0
$378$	−6.32236e8	−0.602086
$379$	1.05379e9	0.994303	0.497152	−	0.867664i	$-0.334379\pi$
$379$	1.05379e9	0.994303	0.497152	+	0.867664i	$0.334379\pi$
$380$	−4.73878e7	−0.0443021
$381$	6.92383e8	0.641370
$382$	2.57541e8	0.236387
$383$	1.86673e9	1.69779	0.848896	−	0.528559i	$-0.177268\pi$
$383$	1.86673e9	1.69779	0.848896	+	0.528559i	$0.177268\pi$
$384$	6.78484e7	0.0611478
$385$	1.33433e9	1.19166
$386$	7.09198e8	0.627642
$387$	4.58426e8	0.402051
$388$	−5.84964e8	−0.508415
$389$	−1.47943e9	−1.27430	−0.637151	−	0.770739i	$-0.719887\pi$
$389$	−1.47943e9	−1.27430	−0.637151	+	0.770739i	$0.719887\pi$
$390$	0	0
$391$	1.06648e9	0.902268
$392$	1.45696e8	0.122164
$393$	−9.04420e8	−0.751616
$394$	−7.10521e7	−0.0585248
$395$	9.59426e8	0.783289
$396$	−6.27419e8	−0.507720
$397$	−2.25104e9	−1.80558	−0.902789	−	0.430084i	$-0.858484\pi$
$397$	−2.25104e9	−1.80558	−0.902789	+	0.430084i	$0.858484\pi$
$398$	6.32365e8	0.502779
$399$	8.31882e7	0.0655626
$400$	−1.36936e8	−0.106981
$401$	1.87787e9	1.45432	0.727161	−	0.686467i	$-0.240839\pi$
$401$	1.87787e9	1.45432	0.727161	+	0.686467i	$0.240839\pi$
$402$	2.30623e8	0.177056
$403$	0	0
$404$	−3.67891e8	−0.277578
$405$	−2.09045e8	−0.156368
$406$	6.61275e8	0.490389
$407$	−4.29236e9	−3.15584
$408$	−2.70631e8	−0.197273
$409$	6.36851e8	0.460263	0.230132	−	0.973160i	$-0.426084\pi$
$409$	6.36851e8	0.460263	0.230132	+	0.973160i	$0.426084\pi$
$410$	4.89037e8	0.350428
$411$	1.44870e9	1.02927
$412$	5.89927e8	0.415583
$413$	1.88754e9	1.31847
$414$	−5.95481e8	−0.412446
$415$	5.51821e8	0.378992
$416$	0	0
$417$	−7.74673e8	−0.523169
$418$	2.40886e8	0.161322
$419$	−1.55430e9	−1.03225	−0.516125	−	0.856513i	$-0.672626\pi$
$419$	−1.55430e9	−1.03225	−0.516125	+	0.856513i	$0.672626\pi$
$420$	−3.21365e8	−0.211654
$421$	9.46533e8	0.618228	0.309114	−	0.951025i	$-0.399968\pi$
$421$	9.46533e8	0.618228	0.309114	+	0.951025i	$0.399968\pi$
$422$	1.74032e9	1.12729
$423$	7.40817e7	0.0475904
$424$	−1.35118e8	−0.0860863
$425$	5.46203e8	0.345138
$426$	−1.02369e9	−0.641556
$427$	−3.95673e8	−0.245945
$428$	−1.06976e9	−0.659526
$429$	0	0
$430$	6.79923e8	0.412401
$431$	−5.19470e7	−0.0312529	−0.0156264	−	0.999878i	$-0.504974\pi$
$431$	−5.19470e7	−0.0312529	−0.0156264	+	0.999878i	$0.504974\pi$
$432$	4.40923e8	0.263129
$433$	1.56815e9	0.928280	0.464140	−	0.885762i	$-0.346363\pi$
$433$	1.56815e9	0.928280	0.464140	+	0.885762i	$0.346363\pi$
$434$	−1.03411e9	−0.607226
$435$	7.70082e8	0.448564
$436$	−9.14467e8	−0.528403
$437$	2.28624e8	0.131050
$438$	7.19252e8	0.408999
$439$	8.59457e8	0.484840	0.242420	−	0.970171i	$-0.422059\pi$
$439$	8.59457e8	0.484840	0.242420	+	0.970171i	$0.422059\pi$
$440$	−9.30568e8	−0.520791
$441$	3.24487e8	0.180162
$442$	0	0
$443$	3.13741e8	0.171458	0.0857292	−	0.996318i	$-0.472678\pi$
$443$	3.13741e8	0.171458	0.0857292	+	0.996318i	$0.472678\pi$
$444$	1.03378e9	0.560518
$445$	1.66548e9	0.895941
$446$	1.00689e9	0.537417
$447$	1.27201e9	0.673617
$448$	1.92454e8	0.101124
$449$	−8.17430e7	−0.0426175	−0.0213088	−	0.999773i	$-0.506783\pi$
$449$	−8.17430e7	−0.0426175	−0.0213088	+	0.999773i	$0.506783\pi$
$450$	−3.04978e8	−0.157770
$451$	−2.48591e9	−1.27605
$452$	−1.40578e9	−0.716032
$453$	1.76375e9	0.891441
$454$	−2.11313e9	−1.05981
$455$	0	0
$456$	−5.80156e7	−0.0286528
$457$	1.94892e9	0.955183	0.477592	−	0.878582i	$-0.341510\pi$
$457$	1.94892e9	0.955183	0.477592	+	0.878582i	$0.341510\pi$
$458$	−5.10490e8	−0.248289
$459$	−1.75873e9	−0.848899
$460$	−8.83199e8	−0.423064
$461$	−3.46407e9	−1.64677	−0.823386	−	0.567481i	$-0.807918\pi$
$461$	−3.46407e9	−1.64677	−0.823386	+	0.567481i	$0.807918\pi$
$462$	1.63359e9	0.770719
$463$	−1.13332e9	−0.530665	−0.265332	−	0.964157i	$-0.585482\pi$
$463$	−1.13332e9	−0.530665	−0.265332	+	0.964157i	$0.585482\pi$
$464$	−4.61174e8	−0.214315
$465$	−1.20426e9	−0.555436
$466$	−1.26655e9	−0.579790
$467$	−2.08755e9	−0.948479	−0.474239	−	0.880396i	$-0.657277\pi$
$467$	−2.08755e9	−0.948479	−0.474239	+	0.880396i	$0.657277\pi$
$468$	0	0
$469$	6.54167e8	0.292809
$470$	1.09876e8	0.0488156
$471$	2.12936e9	0.939024
$472$	−1.31637e9	−0.576212
$473$	−3.45624e9	−1.50172
$474$	1.17460e9	0.506600
$475$	1.17090e8	0.0501295
$476$	−7.67652e8	−0.326242
$477$	−3.00930e8	−0.126955
$478$	2.23694e9	0.936823
$479$	1.84769e8	0.0768164	0.0384082	−	0.999262i	$-0.487771\pi$
$479$	1.84769e8	0.0768164	0.0384082	+	0.999262i	$0.487771\pi$
$480$	2.24121e8	0.0924991
$481$	0	0
$482$	−1.81040e9	−0.736394
$483$	1.55043e9	0.626092
$484$	3.48316e9	1.39641
$485$	−1.93228e9	−0.769086
$486$	1.62747e9	0.643110
$487$	2.28011e9	0.894551	0.447275	−	0.894396i	$-0.352394\pi$
$487$	2.28011e9	0.894551	0.447275	+	0.894396i	$0.352394\pi$
$488$	2.75943e8	0.107485
$489$	−2.10991e9	−0.815985
$490$	4.81269e8	0.184800
$491$	1.95146e9	0.744003	0.372001	−	0.928232i	$-0.378672\pi$
$491$	1.95146e9	0.744003	0.372001	+	0.928232i	$0.378672\pi$
$492$	5.98715e8	0.226643
$493$	1.83951e9	0.691415
$494$	0	0
$495$	−2.07252e9	−0.768035
$496$	7.21187e8	0.265376
$497$	−2.90372e9	−1.06098
$498$	6.75579e8	0.245117
$499$	−3.18870e9	−1.14885	−0.574423	−	0.818559i	$-0.694773\pi$
$499$	−3.18870e9	−1.14885	−0.574423	+	0.818559i	$0.694773\pi$
$500$	−1.50937e9	−0.540010
$501$	−2.04732e9	−0.727369
$502$	−1.73151e9	−0.610887
$503$	−1.77625e9	−0.622322	−0.311161	−	0.950357i	$-0.600718\pi$
$503$	−1.77625e9	−0.622322	−0.311161	+	0.950357i	$0.600718\pi$
$504$	4.28626e8	0.149132
$505$	−1.21524e9	−0.419896
$506$	4.48955e9	1.54055
$507$	0	0
$508$	−1.36967e9	−0.463546
$509$	−5.90283e9	−1.98403	−0.992015	−	0.126117i	$-0.959748\pi$
$509$	−5.90283e9	−1.98403	−0.992015	+	0.126117i	$0.959748\pi$
$510$	−8.93963e8	−0.298417
$511$	2.04018e9	0.676387
$512$	−1.34218e8	−0.0441942
$513$	−3.77023e8	−0.123298
$514$	−1.94861e9	−0.632927
$515$	1.94868e9	0.628658
$516$	8.32411e8	0.266725
$517$	−5.58528e8	−0.177758
$518$	2.93236e9	0.926964
$519$	2.48843e9	0.781340
$520$	0	0
$521$	2.84267e9	0.880633	0.440317	−	0.897843i	$-0.354866\pi$
$521$	2.84267e9	0.880633	0.440317	+	0.897843i	$0.354866\pi$
$522$	−1.02711e9	−0.316060
$523$	−3.06905e9	−0.938096	−0.469048	−	0.883173i	$-0.655403\pi$
$523$	−3.06905e9	−0.938096	−0.469048	+	0.883173i	$0.655403\pi$
$524$	1.78912e9	0.543226
$525$	7.94060e8	0.239495
$526$	4.03603e7	0.0120922
$527$	−2.87664e9	−0.856147
$528$	−1.13927e9	−0.336827
$529$	8.56187e8	0.251463
$530$	−4.46330e8	−0.130224
$531$	−2.93178e9	−0.849767
$532$	−1.64563e8	−0.0473850
$533$	0	0
$534$	2.03900e9	0.579459
$535$	−3.53368e9	−0.997675
$536$	−4.56218e8	−0.127966
$537$	2.79764e9	0.779617
$538$	3.72295e9	1.03074
$539$	−2.44643e9	−0.672933
$540$	1.45648e9	0.398040
$541$	−2.24439e9	−0.609407	−0.304703	−	0.952447i	$-0.598557\pi$
$541$	−2.24439e9	−0.609407	−0.304703	+	0.952447i	$0.598557\pi$
$542$	−2.65419e9	−0.716036
$543$	−2.72561e9	−0.730573
$544$	5.35362e8	0.142578
$545$	−3.02072e9	−0.799323
$546$	0	0
$547$	−3.05066e9	−0.796963	−0.398481	−	0.917176i	$-0.630463\pi$
$547$	−3.05066e9	−0.796963	−0.398481	+	0.917176i	$0.630463\pi$
$548$	−2.86581e9	−0.743902
$549$	6.14569e8	0.158514
$550$	2.29934e9	0.589295
$551$	3.94339e8	0.100424
$552$	−1.08128e9	−0.273621
$553$	3.33178e9	0.837797
$554$	3.26827e9	0.816646
$555$	3.41485e9	0.847904
$556$	1.53246e9	0.378117
$557$	−1.98131e9	−0.485801	−0.242901	−	0.970051i	$-0.578099\pi$
$557$	−1.98131e9	−0.485801	−0.242901	+	0.970051i	$0.578099\pi$
$558$	1.60620e9	0.391363
$559$	0	0
$560$	6.35724e8	0.152972
$561$	4.54426e9	1.08666
$562$	−5.01129e9	−1.19089
$563$	6.38984e9	1.50907	0.754537	−	0.656257i	$-0.227861\pi$
$563$	6.38984e9	1.50907	0.754537	+	0.656257i	$0.227861\pi$
$564$	1.34518e8	0.0315720
$565$	−4.64365e9	−1.08315
$566$	2.19896e8	0.0509754
$567$	−7.25947e8	−0.167249
$568$	2.02506e9	0.463681
$569$	−3.95616e9	−0.900286	−0.450143	−	0.892956i	$-0.648627\pi$
$569$	−3.95616e9	−0.900286	−0.450143	+	0.892956i	$0.648627\pi$
$570$	−1.91640e8	−0.0433436
$571$	1.72462e9	0.387675	0.193837	−	0.981034i	$-0.437907\pi$
$571$	1.72462e9	0.387675	0.193837	+	0.981034i	$0.437907\pi$
$572$	0	0
$573$	1.04152e9	0.231273
$574$	1.69827e9	0.374814
$575$	2.18229e9	0.478713
$576$	−2.98924e8	−0.0651752
$577$	−1.44292e9	−0.312698	−0.156349	−	0.987702i	$-0.549973\pi$
$577$	−1.44292e9	−0.312698	−0.156349	+	0.987702i	$0.549973\pi$
$578$	1.14728e9	0.247128
$579$	2.86805e9	0.614062
$580$	−1.52338e9	−0.324197
$581$	1.91630e9	0.405366
$582$	−2.36564e9	−0.497415
$583$	2.26882e9	0.474199
$584$	−1.42282e9	−0.295601
$585$	0	0
$586$	1.48047e9	0.303919
$587$	−5.41113e9	−1.10422	−0.552108	−	0.833772i	$-0.686177\pi$
$587$	−5.41113e9	−1.10422	−0.552108	+	0.833772i	$0.686177\pi$
$588$	5.89205e8	0.119521
$589$	−6.16670e8	−0.124351
$590$	−4.34832e9	−0.871644
$591$	−2.87341e8	−0.0572586
$592$	−2.04503e9	−0.405111
$593$	1.30921e9	0.257820	0.128910	−	0.991656i	$-0.458852\pi$
$593$	1.30921e9	0.257820	0.128910	+	0.991656i	$0.458852\pi$
$594$	−7.40370e9	−1.44943
$595$	−2.53575e9	−0.493511
$596$	−2.51628e9	−0.486852
$597$	2.55734e9	0.491901
$598$	0	0
$599$	−5.82086e9	−1.10661	−0.553303	−	0.832980i	$-0.686633\pi$
$599$	−5.82086e9	−1.10661	−0.553303	+	0.832980i	$0.686633\pi$
$600$	−5.53779e8	−0.104666
$601$	−9.30512e9	−1.74848	−0.874241	−	0.485492i	$-0.838641\pi$
$601$	−9.30512e9	−1.74848	−0.874241	+	0.485492i	$0.838641\pi$
$602$	2.36116e9	0.441100
$603$	−1.01607e9	−0.188718
$604$	−3.48905e9	−0.644284
$605$	1.15057e10	2.11238
$606$	−1.48778e9	−0.271572
$607$	−3.54931e9	−0.644145	−0.322072	−	0.946715i	$-0.604379\pi$
$607$	−3.54931e9	−0.644145	−0.322072	+	0.946715i	$0.604379\pi$
$608$	1.14766e8	0.0207087
$609$	2.67425e9	0.479779
$610$	9.11509e8	0.162595
$611$	0	0
$612$	1.19234e9	0.210266
$613$	−6.21892e9	−1.09044	−0.545222	−	0.838292i	$-0.683555\pi$
$613$	−6.21892e9	−1.09044	−0.545222	+	0.838292i	$0.683555\pi$
$614$	1.66771e8	0.0290758
$615$	1.97771e9	0.342846
$616$	−3.23156e9	−0.557032
$617$	−3.25505e9	−0.557905	−0.278952	−	0.960305i	$-0.589987\pi$
$617$	−3.25505e9	−0.557905	−0.278952	+	0.960305i	$0.589987\pi$
$618$	2.38571e9	0.406591
$619$	−8.80728e9	−1.49254	−0.746268	−	0.665646i	$-0.768156\pi$
$619$	−8.80728e9	−1.49254	−0.746268	+	0.665646i	$0.768156\pi$
$620$	2.38226e9	0.401438
$621$	−7.02682e9	−1.17744
$622$	5.70470e8	0.0950532
$623$	5.78368e9	0.958289
$624$	0	0
$625$	−2.37401e9	−0.388957
$626$	−2.55230e9	−0.415835
$627$	9.74161e8	0.157832
$628$	−4.21230e9	−0.678674
$629$	8.15714e9	1.30695
$630$	1.41586e9	0.225594
$631$	4.89230e9	0.775194	0.387597	−	0.921829i	$-0.373305\pi$
$631$	4.89230e9	0.775194	0.387597	+	0.921829i	$0.373305\pi$
$632$	−2.32359e9	−0.366142
$633$	7.03800e9	1.10290
$634$	−2.83194e9	−0.441339
$635$	−4.52437e9	−0.701213
$636$	−5.46430e8	−0.0842237
$637$	0	0
$638$	7.74375e9	1.18053
$639$	4.51014e9	0.683812
$640$	−4.43355e8	−0.0668532
$641$	2.90582e9	0.435778	0.217889	−	0.975974i	$-0.430083\pi$
$641$	2.90582e9	0.435778	0.217889	+	0.975974i	$0.430083\pi$
$642$	−4.32618e9	−0.645257
$643$	3.76445e8	0.0558422	0.0279211	−	0.999610i	$-0.491111\pi$
$643$	3.76445e8	0.0558422	0.0279211	+	0.999610i	$0.491111\pi$
$644$	−3.06707e9	−0.452504
$645$	2.74966e9	0.403479
$646$	−4.57775e8	−0.0668096
$647$	3.39531e9	0.492850	0.246425	−	0.969162i	$-0.420744\pi$
$647$	3.39531e9	0.492850	0.246425	+	0.969162i	$0.420744\pi$
$648$	5.06277e8	0.0730930
$649$	2.21037e10	3.17401
$650$	0	0
$651$	−4.18201e9	−0.594089
$652$	4.17382e9	0.589748
$653$	1.29254e10	1.81655	0.908273	−	0.418378i	$-0.137401\pi$
$653$	1.29254e10	1.81655	0.908273	+	0.418378i	$0.137401\pi$
$654$	−3.69818e9	−0.516971
$655$	5.90993e9	0.821746
$656$	−1.18438e9	−0.163805
$657$	−3.16886e9	−0.435937
$658$	3.81563e8	0.0522126
$659$	5.00014e9	0.680586	0.340293	−	0.940319i	$-0.389474\pi$
$659$	5.00014e9	0.680586	0.340293	+	0.940319i	$0.389474\pi$
$660$	−3.76329e9	−0.509523
$661$	−1.27889e10	−1.72238	−0.861189	−	0.508285i	$-0.830279\pi$
$661$	−1.27889e10	−1.72238	−0.861189	+	0.508285i	$0.830279\pi$
$662$	8.07076e9	1.08121
$663$	0	0
$664$	−1.33643e9	−0.177157
$665$	−5.43593e8	−0.0716800
$666$	−4.55461e9	−0.597436
$667$	7.34956e9	0.959006
$668$	4.05002e9	0.525701
$669$	4.07196e9	0.525789
$670$	−1.50700e9	−0.193576
$671$	−4.63346e9	−0.592074
$672$	7.78299e8	0.0989360
$673$	3.92175e9	0.495937	0.247969	−	0.968768i	$-0.420237\pi$
$673$	3.92175e9	0.495937	0.247969	+	0.968768i	$0.420237\pi$
$674$	5.48422e9	0.689931
$675$	−3.59881e9	−0.450397
$676$	0	0
$677$	1.10745e10	1.37171	0.685855	−	0.727739i	$-0.259429\pi$
$677$	1.10745e10	1.37171	0.685855	+	0.727739i	$0.259429\pi$
$678$	−5.68509e9	−0.700540
$679$	−6.71021e9	−0.822606
$680$	1.76844e9	0.215679
$681$	−8.54565e9	−1.03688
$682$	−1.21097e10	−1.46180
$683$	2.65815e9	0.319232	0.159616	−	0.987179i	$-0.448974\pi$
$683$	2.65815e9	0.319232	0.159616	+	0.987179i	$0.448974\pi$
$684$	2.55603e8	0.0305401
$685$	−9.46650e9	−1.12531
$686$	6.50815e9	0.769703
$687$	−2.06446e9	−0.242917
$688$	−1.64667e9	−0.192774
$689$	0	0
$690$	−3.57173e9	−0.413910
$691$	3.23306e9	0.372770	0.186385	−	0.982477i	$-0.440323\pi$
$691$	3.23306e9	0.372770	0.186385	+	0.982477i	$0.440323\pi$
$692$	−4.92261e9	−0.564709
$693$	−7.19721e9	−0.821482
$694$	3.98637e9	0.452709
$695$	5.06210e9	0.571984
$696$	−1.86503e9	−0.209678
$697$	4.72419e9	0.528461
$698$	−2.24787e9	−0.250194
$699$	−5.12202e9	−0.567246
$700$	−1.57081e9	−0.173093
$701$	1.01439e10	1.11223	0.556114	−	0.831106i	$-0.312292\pi$
$701$	1.01439e10	1.11223	0.556114	+	0.831106i	$0.312292\pi$
$702$	0	0
$703$	1.74866e9	0.189828
$704$	2.25370e9	0.243440
$705$	4.44346e8	0.0477594
$706$	−5.27866e9	−0.564556
$707$	−4.22013e9	−0.449116
$708$	−5.32352e9	−0.563745
$709$	−2.65037e8	−0.0279283	−0.0139642	−	0.999902i	$-0.504445\pi$
$709$	−2.65037e8	−0.0279283	−0.0139642	+	0.999902i	$0.504445\pi$
$710$	6.68929e9	0.701417
$711$	−5.17501e9	−0.539967
$712$	−4.03355e9	−0.418801
$713$	−1.14933e10	−1.18749
$714$	−3.10445e9	−0.319184
$715$	0	0
$716$	−5.53429e9	−0.563464
$717$	9.04639e9	0.916554
$718$	2.98378e8	0.0300837
$719$	2.61758e9	0.262633	0.131317	−	0.991340i	$-0.458080\pi$
$719$	2.61758e9	0.262633	0.131317	+	0.991340i	$0.458080\pi$
$720$	−9.87423e8	−0.0985915
$721$	6.76714e9	0.672406
$722$	7.05284e9	0.697403
$723$	−7.32142e9	−0.720462
$724$	5.39179e9	0.528017
$725$	3.76410e9	0.366842
$726$	1.40862e10	1.36620
$727$	−1.98815e10	−1.91902	−0.959509	−	0.281676i	$-0.909110\pi$
$727$	−1.98815e10	−1.91902	−0.959509	+	0.281676i	$0.909110\pi$
$728$	0	0
$729$	8.74416e9	0.835934
$730$	−4.69995e9	−0.447160
$731$	6.56819e9	0.621920
$732$	1.11594e9	0.105160
$733$	−1.77576e10	−1.66541	−0.832703	−	0.553720i	$-0.813208\pi$
$733$	−1.77576e10	−1.66541	−0.832703	+	0.553720i	$0.813208\pi$
$734$	5.76388e9	0.537996
$735$	1.94629e9	0.180802
$736$	2.13898e9	0.197758
$737$	7.66052e9	0.704891
$738$	−2.63780e9	−0.241571
$739$	1.12945e10	1.02947	0.514734	−	0.857350i	$-0.327891\pi$
$739$	1.12945e10	1.02947	0.514734	+	0.857350i	$0.327891\pi$
$740$	−6.75526e9	−0.612817
$741$	0	0
$742$	−1.54996e9	−0.139286
$743$	−7.83255e9	−0.700555	−0.350278	−	0.936646i	$-0.613913\pi$
$743$	−7.83255e9	−0.700555	−0.350278	+	0.936646i	$0.613913\pi$
$744$	2.91654e9	0.259635
$745$	−8.31192e9	−0.736469
$746$	2.32803e9	0.205307
$747$	−2.97645e9	−0.261262
$748$	−8.98946e9	−0.785377
$749$	−1.22713e10	−1.06710
$750$	−6.10403e9	−0.528327
$751$	−2.35874e9	−0.203208	−0.101604	−	0.994825i	$-0.532397\pi$
$751$	−2.35874e9	−0.203208	−0.101604	+	0.994825i	$0.532397\pi$
$752$	−2.66103e8	−0.0228185
$753$	−7.00235e9	−0.597670
$754$	0	0
$755$	−1.15252e10	−0.974618
$756$	5.05789e9	0.425739
$757$	1.28454e10	1.07625	0.538125	−	0.842865i	$-0.319133\pi$
$757$	1.28454e10	1.07625	0.538125	+	0.842865i	$0.319133\pi$
$758$	−8.43036e9	−0.703079
$759$	1.81561e10	1.50722
$760$	3.79103e8	0.0313263
$761$	9.88665e8	0.0813210	0.0406605	−	0.999173i	$-0.487054\pi$
$761$	9.88665e8	0.0813210	0.0406605	+	0.999173i	$0.487054\pi$
$762$	−5.53906e9	−0.453517
$763$	−1.04900e10	−0.854947
$764$	−2.06033e9	−0.167151
$765$	3.93859e9	0.318072
$766$	−1.49338e10	−1.20052
$767$	0	0
$768$	−5.42787e8	−0.0432380
$769$	−1.33874e10	−1.06158	−0.530792	−	0.847502i	$-0.678106\pi$
$769$	−1.33874e10	−1.06158	−0.530792	+	0.847502i	$0.678106\pi$
$770$	−1.06747e10	−0.842631
$771$	−7.88033e9	−0.619233
$772$	−5.67358e9	−0.443810
$773$	1.06787e10	0.831551	0.415776	−	0.909467i	$-0.363510\pi$
$773$	1.06787e10	0.831551	0.415776	+	0.909467i	$0.363510\pi$
$774$	−3.66741e9	−0.284293
$775$	−5.88633e9	−0.454243
$776$	4.67971e9	0.359503
$777$	1.18587e10	0.906908
$778$	1.18355e10	0.901067
$779$	1.01273e9	0.0767562
$780$	0	0
$781$	−3.40036e10	−2.55415
$782$	−8.53187e9	−0.638000
$783$	−1.21201e10	−0.902281
$784$	−1.16556e9	−0.0863833
$785$	−1.39143e10	−1.02664
$786$	7.23536e9	0.531473
$787$	−1.54726e10	−1.13150	−0.565748	−	0.824579i	$-0.691412\pi$
$787$	−1.54726e10	−1.13150	−0.565748	+	0.824579i	$0.691412\pi$
$788$	5.68417e8	0.0413833
$789$	1.63220e8	0.0118305
$790$	−7.67541e9	−0.553869
$791$	−1.61259e10	−1.15853
$792$	5.01935e9	0.359012
$793$	0	0
$794$	1.80083e10	1.27674
$795$	−1.80500e9	−0.127406
$796$	−5.05892e9	−0.355519
$797$	1.83143e10	1.28141	0.640703	−	0.767789i	$-0.278643\pi$
$797$	1.83143e10	1.28141	0.640703	+	0.767789i	$0.278643\pi$
$798$	−6.65505e8	−0.0463598
$799$	1.06142e9	0.0736162
$800$	1.09548e9	0.0756470
$801$	−8.98336e9	−0.617625
$802$	−1.50230e10	−1.02836
$803$	2.38912e10	1.62829
$804$	−1.84498e9	−0.125198
$805$	−1.01313e10	−0.684510
$806$	0	0
$807$	1.50559e10	1.00844
$808$	2.94313e9	0.196277
$809$	6.06212e9	0.402536	0.201268	−	0.979536i	$-0.435494\pi$
$809$	6.06212e9	0.402536	0.201268	+	0.979536i	$0.435494\pi$
$810$	1.67236e9	0.110569
$811$	2.18462e10	1.43815	0.719074	−	0.694933i	$-0.244566\pi$
$811$	2.18462e10	1.43815	0.719074	+	0.694933i	$0.244566\pi$
$812$	−5.29020e9	−0.346758
$813$	−1.07338e10	−0.700544
$814$	3.43389e10	2.23152
$815$	1.37872e10	0.892121
$816$	2.16505e9	0.139493
$817$	1.40803e9	0.0903307
$818$	−5.09481e9	−0.325455
$819$	0	0
$820$	−3.91230e9	−0.247790
$821$	2.61422e10	1.64870	0.824350	−	0.566080i	$-0.191541\pi$
$821$	2.61422e10	1.64870	0.824350	+	0.566080i	$0.191541\pi$
$822$	−1.15896e10	−0.727807
$823$	8.16231e9	0.510403	0.255202	−	0.966888i	$-0.417858\pi$
$823$	8.16231e9	0.510403	0.255202	+	0.966888i	$0.417858\pi$
$824$	−4.71941e9	−0.293862
$825$	9.29870e9	0.576545
$826$	−1.51003e10	−0.932301
$827$	2.10648e10	1.29505	0.647527	−	0.762042i	$-0.275803\pi$
$827$	2.10648e10	1.29505	0.647527	+	0.762042i	$0.275803\pi$
$828$	4.76385e9	0.291643
$829$	2.88820e10	1.76071	0.880353	−	0.474319i	$-0.157306\pi$
$829$	2.88820e10	1.76071	0.880353	+	0.474319i	$0.157306\pi$
$830$	−4.41457e9	−0.267988
$831$	1.32172e10	0.798977
$832$	0	0
$833$	4.64915e9	0.278687
$834$	6.19738e9	0.369936
$835$	1.33782e10	0.795236
$836$	−1.92708e9	−0.114072
$837$	1.89535e10	1.11725
$838$	1.24344e10	0.729911
$839$	1.18330e10	0.691719	0.345859	−	0.938286i	$-0.387587\pi$
$839$	1.18330e10	0.691719	0.345859	+	0.938286i	$0.387587\pi$
$840$	2.57092e9	0.149662
$841$	−4.57306e9	−0.265107
$842$	−7.57226e9	−0.437153
$843$	−2.02661e10	−1.16513
$844$	−1.39226e10	−0.797114
$845$	0	0
$846$	−5.92653e8	−0.0336515
$847$	3.99558e10	2.25937
$848$	1.08095e9	0.0608722
$849$	8.89278e8	0.0498725
$850$	−4.36962e9	−0.244049
$851$	3.25909e10	1.81277
$852$	8.18951e9	0.453649
$853$	4.54152e9	0.250542	0.125271	−	0.992123i	$-0.460020\pi$
$853$	4.54152e9	0.250542	0.125271	+	0.992123i	$0.460020\pi$
$854$	3.16538e9	0.173910
$855$	8.44322e8	0.0461984
$856$	8.55806e9	0.466355
$857$	−2.71958e10	−1.47594	−0.737971	−	0.674833i	$-0.764216\pi$
$857$	−2.71958e10	−1.47594	−0.737971	+	0.674833i	$0.764216\pi$
$858$	0	0
$859$	−1.24708e10	−0.671304	−0.335652	−	0.941986i	$-0.608957\pi$
$859$	−1.24708e10	−0.671304	−0.335652	+	0.941986i	$0.608957\pi$
$860$	−5.43938e9	−0.291612
$861$	6.86794e9	0.366704
$862$	4.15576e8	0.0220991
$863$	−3.88024e9	−0.205504	−0.102752	−	0.994707i	$-0.532765\pi$
$863$	−3.88024e9	−0.205504	−0.102752	+	0.994707i	$0.532765\pi$
$864$	−3.52738e9	−0.186061
$865$	−1.62606e10	−0.854243
$866$	−1.25452e10	−0.656393
$867$	4.63969e9	0.241781
$868$	8.27284e9	0.429374
$869$	3.90163e10	2.01686
$870$	−6.16066e9	−0.317183
$871$	0	0
$872$	7.31574e9	0.373638
$873$	1.04225e10	0.530177
$874$	−1.82899e9	−0.0926662
$875$	−1.73143e10	−0.873727
$876$	−5.75402e9	−0.289206
$877$	−3.12728e10	−1.56555	−0.782777	−	0.622302i	$-0.786198\pi$
$877$	−3.12728e10	−1.56555	−0.782777	+	0.622302i	$0.786198\pi$
$878$	−6.87566e9	−0.342834
$879$	5.98715e9	0.297344
$880$	7.44454e9	0.368255
$881$	8.00605e9	0.394460	0.197230	−	0.980357i	$-0.436805\pi$
$881$	8.00605e9	0.394460	0.197230	+	0.980357i	$0.436805\pi$
$882$	−2.59590e9	−0.127394
$883$	−5.19038e9	−0.253709	−0.126855	−	0.991921i	$-0.540488\pi$
$883$	−5.19038e9	−0.253709	−0.126855	+	0.991921i	$0.540488\pi$
$884$	0	0
$885$	−1.75850e10	−0.852785
$886$	−2.50993e9	−0.121239
$887$	2.60354e10	1.25266	0.626329	−	0.779559i	$-0.284557\pi$
$887$	2.60354e10	1.25266	0.626329	+	0.779559i	$0.284557\pi$
$888$	−8.27028e9	−0.396346
$889$	−1.57117e10	−0.750010
$890$	−1.33238e10	−0.633526
$891$	−8.50108e9	−0.402626
$892$	−8.05514e9	−0.380011
$893$	2.27538e8	0.0106924
$894$	−1.01760e10	−0.476319
$895$	−1.82812e10	−0.852360
$896$	−1.53963e9	−0.0715054
$897$	0	0
$898$	6.53944e8	0.0301351
$899$	−1.98241e10	−0.909985
$900$	2.43982e9	0.111560
$901$	−4.31164e9	−0.196384
$902$	1.98873e10	0.902304
$903$	9.54871e9	0.431556
$904$	1.12462e10	0.506311
$905$	1.78105e10	0.798740
$906$	−1.41100e10	−0.630344
$907$	−3.66202e10	−1.62965	−0.814826	−	0.579706i	$-0.803167\pi$
$907$	−3.66202e10	−1.62965	−0.814826	+	0.579706i	$0.803167\pi$
$908$	1.69050e10	0.749402
$909$	6.55482e9	0.289459
$910$	0	0
$911$	3.91136e10	1.71401	0.857006	−	0.515306i	$-0.172322\pi$
$911$	3.91136e10	1.71401	0.857006	+	0.515306i	$0.172322\pi$
$912$	4.64125e8	0.0202606
$913$	2.24405e10	0.975854
$914$	−1.55913e10	−0.675416
$915$	3.68622e9	0.159077
$916$	4.08392e9	0.175567
$917$	2.05233e10	0.878930
$918$	1.40699e10	0.600262
$919$	1.29642e10	0.550986	0.275493	−	0.961303i	$-0.411159\pi$
$919$	1.29642e10	0.550986	0.275493	+	0.961303i	$0.411159\pi$
$920$	7.06559e9	0.299151
$921$	6.74436e8	0.0284467
$922$	2.77126e10	1.16444
$923$	0	0
$924$	−1.30687e10	−0.544980
$925$	1.66915e10	0.693427
$926$	9.06659e9	0.375237
$927$	−1.05109e10	−0.433372
$928$	3.68939e9	0.151543
$929$	5.89433e9	0.241201	0.120601	−	0.992701i	$-0.461518\pi$
$929$	5.89433e9	0.241201	0.120601	+	0.992701i	$0.461518\pi$
$930$	9.63407e9	0.392753
$931$	9.96646e8	0.0404778
$932$	1.01324e10	0.409974
$933$	2.30703e9	0.0929966
$934$	1.67004e10	0.670676
$935$	−2.96945e10	−1.18805
$936$	0	0
$937$	2.40381e10	0.954577	0.477288	−	0.878747i	$-0.341620\pi$
$937$	2.40381e10	0.954577	0.477288	+	0.878747i	$0.341620\pi$
$938$	−5.23334e9	−0.207047
$939$	−1.03217e10	−0.406838
$940$	−8.79005e8	−0.0345178
$941$	−5.30161e9	−0.207417	−0.103708	−	0.994608i	$-0.533071\pi$
$941$	−5.30161e9	−0.207417	−0.103708	+	0.994608i	$0.533071\pi$
$942$	−1.70349e10	−0.663990
$943$	1.88750e10	0.732986
$944$	1.05310e10	0.407443
$945$	1.67075e10	0.644021
$946$	2.76499e10	1.06188
$947$	2.87596e10	1.10042	0.550210	−	0.835027i	$-0.314548\pi$
$947$	2.87596e10	1.10042	0.550210	+	0.835027i	$0.314548\pi$
$948$	−9.39679e9	−0.358220
$949$	0	0
$950$	−9.36723e8	−0.0354469
$951$	−1.14526e10	−0.431790
$952$	6.14122e9	0.230688
$953$	−3.46970e10	−1.29858	−0.649288	−	0.760543i	$-0.724933\pi$
$953$	−3.46970e10	−1.29858	−0.649288	+	0.760543i	$0.724933\pi$
$954$	2.40744e9	0.0897711
$955$	−6.80578e9	−0.252852
$956$	−1.78956e10	−0.662434
$957$	3.13163e10	1.15499
$958$	−1.47815e9	−0.0543174
$959$	−3.28741e10	−1.20362
$960$	−1.79296e9	−0.0654067
$961$	3.48839e9	0.126792
$962$	0	0
$963$	1.90602e10	0.687756
$964$	1.44832e10	0.520710
$965$	−1.87413e10	−0.671358
$966$	−1.24035e10	−0.442714
$967$	2.32799e10	0.827921	0.413960	−	0.910295i	$-0.364145\pi$
$967$	2.32799e10	0.827921	0.413960	+	0.910295i	$0.364145\pi$
$968$	−2.78652e10	−0.987414
$969$	−1.85128e9	−0.0653641
$970$	1.54583e10	0.543826
$971$	−2.08019e9	−0.0729181	−0.0364591	−	0.999335i	$-0.511608\pi$
$971$	−2.08019e9	−0.0729181	−0.0364591	+	0.999335i	$0.511608\pi$
$972$	−1.30197e10	−0.454747
$973$	1.75790e10	0.611787
$974$	−1.82409e10	−0.632543
$975$	0	0
$976$	−2.20754e9	−0.0760037
$977$	−9.82633e9	−0.337101	−0.168551	−	0.985693i	$-0.553909\pi$
$977$	−9.82633e9	−0.337101	−0.168551	+	0.985693i	$0.553909\pi$
$978$	1.68793e10	0.576989
$979$	6.77288e10	2.30693
$980$	−3.85015e9	−0.130673
$981$	1.62933e10	0.551021
$982$	−1.56117e10	−0.526089
$983$	3.81722e10	1.28177	0.640885	−	0.767637i	$-0.278568\pi$
$983$	3.81722e10	1.28177	0.640885	+	0.767637i	$0.278568\pi$
$984$	−4.78972e9	−0.160261
$985$	1.87763e9	0.0626011
$986$	−1.47161e10	−0.488904
$987$	1.54307e9	0.0510830
$988$	0	0
$989$	2.62424e10	0.862615
$990$	1.65802e10	0.543083
$991$	1.33232e10	0.434860	0.217430	−	0.976076i	$-0.430233\pi$
$991$	1.33232e10	0.434860	0.217430	+	0.976076i	$0.430233\pi$
$992$	−5.76949e9	−0.187649
$993$	3.26388e10	1.05782
$994$	2.32298e10	0.750227
$995$	−1.67109e10	−0.537798
$996$	−5.40464e9	−0.173324
$997$	3.83592e10	1.22585	0.612923	−	0.790142i	$-0.289993\pi$
$997$	3.83592e10	1.22585	0.612923	+	0.790142i	$0.289993\pi$
$998$	2.55096e10	0.812356
$999$	−5.37456e10	−1.70555

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	338.8.a.m.1.3		8
13.2	odd	12		26.8.e.a.17.3	✓	16
13.5	odd	4		338.8.b.i.337.11		16
13.7	odd	12		26.8.e.a.23.3	yes	16
13.8	odd	4		338.8.b.i.337.3		16
13.12	even	2		338.8.a.n.1.3		8
39.2	even	12		234.8.l.c.199.7		16
39.20	even	12		234.8.l.c.127.6		16
52.7	even	12		208.8.w.b.49.3		16
52.15	even	12		208.8.w.b.17.3		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
26.8.e.a.17.3	✓	16	13.2	odd	12
26.8.e.a.23.3	yes	16	13.7	odd	12
208.8.w.b.17.3		16	52.15	even	12
208.8.w.b.49.3		16	52.7	even	12
234.8.l.c.127.6		16	39.20	even	12
234.8.l.c.199.7		16	39.2	even	12
338.8.a.m.1.3		8	1.1	even	1	trivial
338.8.a.n.1.3		8	13.12	even	2
338.8.b.i.337.3		16	13.8	odd	4
338.8.b.i.337.11		16	13.5	odd	4

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:	\( N \)	\(=\)	\( 338 = 2 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	338.a (trivial)

\(f(q)\)	\(=\)	\(q-8.00000 q^{2} -32.3527 q^{3} +64.0000 q^{4} +211.408 q^{5} +258.821 q^{6} +734.154 q^{7} -512.000 q^{8} -1140.31 q^{9} -1691.27 q^{10} +8597.18 q^{11} -2070.57 q^{12} -5873.23 q^{14} -6839.62 q^{15} +4096.00 q^{16} -16338.0 q^{17} +9122.45 q^{18} -3502.39 q^{19} +13530.1 q^{20} -23751.8 q^{21} -68777.4 q^{22} -65276.4 q^{23} +16564.6 q^{24} -33431.5 q^{25} +107647. q^{27} +46985.8 q^{28} -112591. q^{29} +54716.9 q^{30} +176071. q^{31} -32768.0 q^{32} -278142. q^{33} +130704. q^{34} +155206. q^{35} -72979.6 q^{36} -499275. q^{37} +28019.1 q^{38} -108241. q^{40} -289155. q^{41} +190015. q^{42} -402020. q^{43} +550219. q^{44} -241070. q^{45} +522211. q^{46} -64966.5 q^{47} -132516. q^{48} -284562. q^{49} +267452. q^{50} +528576. q^{51} +263903. q^{53} -861177. q^{54} +1.81751e6 q^{55} -375887. q^{56} +113312. q^{57} +900731. q^{58} +2.57104e6 q^{59} -437736. q^{60} -538951. q^{61} -1.40857e6 q^{62} -837160. q^{63} +262144. q^{64} +2.22513e6 q^{66} +891050. q^{67} -1.04563e6 q^{68} +2.11187e6 q^{69} -1.24165e6 q^{70} -3.95520e6 q^{71} +583837. q^{72} +2.77895e6 q^{73} +3.99420e6 q^{74} +1.08160e6 q^{75} -224153. q^{76} +6.31165e6 q^{77} +4.53826e6 q^{79} +865928. q^{80} -988822. q^{81} +2.31324e6 q^{82} +2.61022e6 q^{83} -1.52012e6 q^{84} -3.45398e6 q^{85} +3.21616e6 q^{86} +3.64263e6 q^{87} -4.40176e6 q^{88} +7.87803e6 q^{89} +1.92856e6 q^{90} -4.17769e6 q^{92} -5.69636e6 q^{93} +519732. q^{94} -740435. q^{95} +1.06013e6 q^{96} -9.14006e6 q^{97} +2.27649e6 q^{98} -9.80341e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 64 q^{2} + 512 q^{4} - 304 q^{5} - 2160 q^{7} - 4096 q^{8} + 4684 q^{9} + 2432 q^{10} - 2160 q^{11} + 17280 q^{14} - 21296 q^{15} + 32768 q^{16} + 41520 q^{17} - 37472 q^{18} - 66240 q^{19} - 19456 q^{20}+ \cdots - 44172000 q^{99}+O(q^{100}) \) 8 * q - 64 * q^2 + 512 * q^4 - 304 * q^5 - 2160 * q^7 - 4096 * q^8 + 4684 * q^9 + 2432 * q^10 - 2160 * q^11 + 17280 * q^14 - 21296 * q^15 + 32768 * q^16 + 41520 * q^17 - 37472 * q^18 - 66240 * q^19 - 19456 * q^20 + 51040 * q^21 + 17280 * q^22 - 7560 * q^23 + 136980 * q^25 + 66960 * q^27 - 138240 * q^28 + 346056 * q^29 + 170368 * q^30 + 346320 * q^31 - 262144 * q^32 - 496320 * q^33 - 332160 * q^34 + 283248 * q^35 + 299776 * q^36 - 142800 * q^37 + 529920 * q^38 + 155648 * q^40 - 1525280 * q^41 - 408320 * q^42 + 285360 * q^43 - 138240 * q^44 + 254416 * q^45 + 60480 * q^46 - 1787952 * q^47 - 2131660 * q^49 - 1095840 * q^50 - 478640 * q^51 + 1833300 * q^53 - 535680 * q^54 + 1091472 * q^55 + 1105920 * q^56 + 1694208 * q^57 - 2768448 * q^58 + 1034000 * q^59 - 1362944 * q^60 + 4626000 * q^61 - 2770560 * q^62 - 3577760 * q^63 + 2097152 * q^64 + 3970560 * q^66 - 4959456 * q^67 + 2657280 * q^68 + 18027316 * q^69 - 2265984 * q^70 - 2591040 * q^71 - 2398208 * q^72 - 5390592 * q^73 + 1142400 * q^74 - 10898032 * q^75 - 4239360 * q^76 + 10288740 * q^77 + 17071744 * q^79 - 1245184 * q^80 + 15745864 * q^81 + 12202240 * q^82 - 17216416 * q^83 + 3266560 * q^84 + 19975440 * q^85 - 2282880 * q^86 + 10460440 * q^87 + 1105920 * q^88 + 12984960 * q^89 - 2035328 * q^90 - 483840 * q^92 + 8072768 * q^93 + 14303616 * q^94 + 33837504 * q^95 - 13500864 * q^97 + 17053280 * q^98 - 44172000 * q^99

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