LMFDB - Embedded newform 2368.2.a.bj.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2368,2,Mod(1,2368)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2368, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("2368.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2368 = 2^{6} \cdot 37 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2368.a (trivial)

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$18.9085751986$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 20x^{6} + 116x^{4} - 221x^{2} + 128 $ x^8 - 20x^6 + 116x^4 - 221*x^2 + 128
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 1184)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-3.39115$ of defining polynomial
Character	$\chi$	$=$	2368.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.39115 q^{3} -0.805051 q^{5} +1.94652 q^{7} +8.49990 q^{9} +O(q^{10})$	$q-3.39115 q^{3} -0.805051 q^{5} +1.94652 q^{7} +8.49990 q^{9} -0.661102 q^{11} +5.64790 q^{13} +2.73005 q^{15} +2.00000 q^{17} +2.15142 q^{19} -6.60095 q^{21} +3.59605 q^{23} -4.35189 q^{25} -18.6510 q^{27} +5.64790 q^{29} -6.32610 q^{31} +2.24190 q^{33} -1.56705 q^{35} -1.00000 q^{37} -19.1529 q^{39} +4.95305 q^{41} +5.04068 q^{43} -6.84285 q^{45} -3.51357 q^{47} -3.21105 q^{49} -6.78230 q^{51} -6.39884 q^{53} +0.532221 q^{55} -7.29579 q^{57} +14.3938 q^{59} -3.19495 q^{61} +16.5452 q^{63} -4.54684 q^{65} -13.6928 q^{67} -12.1947 q^{69} -10.2959 q^{71} +13.4999 q^{73} +14.7579 q^{75} -1.28685 q^{77} +3.59605 q^{79} +37.7485 q^{81} +4.83578 q^{83} -1.61010 q^{85} -19.1529 q^{87} +5.51621 q^{89} +10.9938 q^{91} +21.4527 q^{93} -1.73200 q^{95} +2.00000 q^{97} -5.61930 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 2 q^{5} + 16 q^{9}+O(q^{10}) $ 8 * q + 2 * q^5 + 16 * q^9	$ 8 q + 2 q^{5} + 16 q^{9} - 2 q^{13} + 16 q^{17} - 2 q^{21} + 22 q^{25} - 2 q^{29} + 30 q^{33} - 8 q^{37} + 36 q^{41} - 16 q^{45} + 42 q^{49} + 2 q^{53} + 36 q^{57} - 34 q^{61} + 12 q^{65} - 2 q^{69} + 56 q^{73} + 14 q^{77} + 48 q^{81} + 4 q^{85} + 20 q^{89} + 12 q^{93} + 16 q^{97}+O(q^{100}) $ 8 * q + 2 * q^5 + 16 * q^9 - 2 * q^13 + 16 * q^17 - 2 * q^21 + 22 * q^25 - 2 * q^29 + 30 * q^33 - 8 * q^37 + 36 * q^41 - 16 * q^45 + 42 * q^49 + 2 * q^53 + 36 * q^57 - 34 * q^61 + 12 * q^65 - 2 * q^69 + 56 * q^73 + 14 * q^77 + 48 * q^81 + 4 * q^85 + 20 * q^89 + 12 * q^93 + 16 * q^97

Display 100 coefficients 10 coefficients

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
$3$	−3.39115	−1.95788	−0.978941	−	0.204146i	$-0.934558\pi$
$3$	−3.39115	−1.95788	−0.978941	+	0.204146i	$0.934558\pi$
$4$	0	0
$5$	−0.805051	−0.360030	−0.180015	−	0.983664i	$-0.557615\pi$
$5$	−0.805051	−0.360030	−0.180015	+	0.983664i	$0.557615\pi$
$6$	0	0
$7$	1.94652	0.735716	0.367858	−	0.929882i	$-0.380091\pi$
$7$	1.94652	0.735716	0.367858	+	0.929882i	$0.380091\pi$
$8$	0	0
$9$	8.49990	2.83330
$10$	0	0
$11$	−0.661102	−0.199330	−0.0996649	−	0.995021i	$-0.531777\pi$
$11$	−0.661102	−0.199330	−0.0996649	+	0.995021i	$0.531777\pi$
$12$	0	0
$13$	5.64790	1.56644	0.783222	−	0.621742i	$-0.213575\pi$
$13$	5.64790	1.56644	0.783222	+	0.621742i	$0.213575\pi$
$14$	0	0
$15$	2.73005	0.704895
$16$	0	0
$17$	2.00000	0.485071	0.242536	−	0.970143i	$-0.422021\pi$
$17$	2.00000	0.485071	0.242536	+	0.970143i	$0.422021\pi$
$18$	0	0
$19$	2.15142	0.493570	0.246785	−	0.969070i	$-0.420626\pi$
$19$	2.15142	0.493570	0.246785	+	0.969070i	$0.420626\pi$
$20$	0	0
$21$	−6.60095	−1.44045
$22$	0	0
$23$	3.59605	0.749828	0.374914	−	0.927060i	$-0.377672\pi$
$23$	3.59605	0.749828	0.374914	+	0.927060i	$0.377672\pi$
$24$	0	0
$25$	−4.35189	−0.870379
$26$	0	0
$27$	−18.6510	−3.58938
$28$	0	0
$29$	5.64790	1.04879	0.524394	−	0.851476i	$-0.324292\pi$
$29$	5.64790	1.04879	0.524394	+	0.851476i	$0.324292\pi$
$30$	0	0
$31$	−6.32610	−1.13620	−0.568100	−	0.822959i	$-0.692321\pi$
$31$	−6.32610	−1.13620	−0.568100	+	0.822959i	$0.692321\pi$
$32$	0	0
$33$	2.24190	0.390264
$34$	0	0
$35$	−1.56705	−0.264880
$36$	0	0
$37$	−1.00000	−0.164399
$37$	−1.00000	−0.164399
$38$	0	0
$39$	−19.1529	−3.06691
$40$	0	0
$41$	4.95305	0.773537	0.386768	−	0.922177i	$-0.373591\pi$
$41$	4.95305	0.773537	0.386768	+	0.922177i	$0.373591\pi$
$42$	0	0
$43$	5.04068	0.768696	0.384348	−	0.923188i	$-0.374426\pi$
$43$	5.04068	0.768696	0.384348	+	0.923188i	$0.374426\pi$
$44$	0	0
$45$	−6.84285	−1.02007
$46$	0	0
$47$	−3.51357	−0.512507	−0.256254	−	0.966610i	$-0.582488\pi$
$47$	−3.51357	−0.512507	−0.256254	+	0.966610i	$0.582488\pi$
$48$	0	0
$49$	−3.21105	−0.458721
$50$	0	0
$51$	−6.78230	−0.949712
$52$	0	0
$53$	−6.39884	−0.878948	−0.439474	−	0.898255i	$-0.644835\pi$
$53$	−6.39884	−0.878948	−0.439474	+	0.898255i	$0.644835\pi$
$54$	0	0
$55$	0.532221	0.0717647
$56$	0	0
$57$	−7.29579	−0.966351
$58$	0	0
$59$	14.3938	1.87392	0.936958	−	0.349443i	$-0.113629\pi$
$59$	14.3938	1.87392	0.936958	+	0.349443i	$0.113629\pi$
$60$	0	0
$61$	−3.19495	−0.409071	−0.204536	−	0.978859i	$-0.565568\pi$
$61$	−3.19495	−0.409071	−0.204536	+	0.978859i	$0.565568\pi$
$62$	0	0
$63$	16.5452	2.08450
$64$	0	0
$65$	−4.54684	−0.563966
$66$	0	0
$67$	−13.6928	−1.67284	−0.836419	−	0.548091i	$-0.815355\pi$
$67$	−13.6928	−1.67284	−0.836419	+	0.548091i	$0.815355\pi$
$68$	0	0
$69$	−12.1947	−1.46807
$70$	0	0
$71$	−10.2959	−1.22190	−0.610948	−	0.791671i	$-0.709211\pi$
$71$	−10.2959	−1.22190	−0.610948	+	0.791671i	$0.709211\pi$
$72$	0	0
$73$	13.4999	1.58004	0.790022	−	0.613079i	$-0.210069\pi$
$73$	13.4999	1.58004	0.790022	+	0.613079i	$0.210069\pi$
$74$	0	0
$75$	14.7579	1.70410
$76$	0	0
$77$	−1.28685	−0.146650
$78$	0	0
$79$	3.59605	0.404587	0.202294	−	0.979325i	$-0.435160\pi$
$79$	3.59605	0.404587	0.202294	+	0.979325i	$0.435160\pi$
$80$	0	0
$81$	37.7485	4.19428
$82$	0	0
$83$	4.83578	0.530795	0.265398	−	0.964139i	$-0.414497\pi$
$83$	4.83578	0.530795	0.265398	+	0.964139i	$0.414497\pi$
$84$	0	0
$85$	−1.61010	−0.174640
$86$	0	0
$87$	−19.1529	−2.05340
$88$	0	0
$89$	5.51621	0.584717	0.292358	−	0.956309i	$-0.405560\pi$
$89$	5.51621	0.584717	0.292358	+	0.956309i	$0.405560\pi$
$90$	0	0
$91$	10.9938	1.15246
$92$	0	0
$93$	21.4527	2.22455
$94$	0	0
$95$	−1.73200	−0.177700
$96$	0	0
$97$	2.00000	0.203069	0.101535	−	0.994832i	$-0.467625\pi$
$97$	2.00000	0.203069	0.101535	+	0.994832i	$0.467625\pi$
$98$	0	0
$99$	−5.61930	−0.564761
$100$	0	0
$101$	10.9908	1.09363	0.546815	−	0.837253i	$-0.315840\pi$
$101$	10.9908	1.09363	0.546815	+	0.837253i	$0.315840\pi$
$102$	0	0
$103$	−8.97051	−0.883890	−0.441945	−	0.897042i	$-0.645711\pi$
$103$	−8.97051	−0.883890	−0.441945	+	0.897042i	$0.645711\pi$
$104$	0	0
$105$	5.31410	0.518603
$106$	0	0
$107$	16.0130	1.54804	0.774018	−	0.633163i	$-0.218244\pi$
$107$	16.0130	1.54804	0.774018	+	0.633163i	$0.218244\pi$
$108$	0	0
$109$	−15.3897	−1.47406	−0.737032	−	0.675857i	$-0.763774\pi$
$109$	−15.3897	−1.47406	−0.737032	+	0.675857i	$0.763774\pi$
$110$	0	0
$111$	3.39115	0.321874
$112$	0	0
$113$	14.6099	1.37438	0.687191	−	0.726476i	$-0.258843\pi$
$113$	14.6099	1.37438	0.687191	+	0.726476i	$0.258843\pi$
$114$	0	0
$115$	−2.89500	−0.269960
$116$	0	0
$117$	48.0065	4.43821
$118$	0	0
$119$	3.89305	0.356875
$120$	0	0
$121$	−10.5629	−0.960268
$122$	0	0
$123$	−16.7965	−1.51449
$124$	0	0
$125$	7.52875	0.673392
$126$	0	0
$127$	14.1889	1.25906	0.629531	−	0.776975i	$-0.283247\pi$
$127$	14.1889	1.25906	0.629531	+	0.776975i	$0.283247\pi$
$128$	0	0
$129$	−17.0937	−1.50502
$130$	0	0
$131$	−0.829217	−0.0724490	−0.0362245	−	0.999344i	$-0.511533\pi$
$131$	−0.829217	−0.0724490	−0.0362245	+	0.999344i	$0.511533\pi$
$132$	0	0
$133$	4.18779	0.363128
$134$	0	0
$135$	15.0150	1.29228
$136$	0	0
$137$	−4.88064	−0.416981	−0.208491	−	0.978024i	$-0.566855\pi$
$137$	−4.88064	−0.416981	−0.208491	+	0.978024i	$0.566855\pi$
$138$	0	0
$139$	1.03483	0.0877730	0.0438865	−	0.999037i	$-0.486026\pi$
$139$	1.03483	0.0877730	0.0438865	+	0.999037i	$0.486026\pi$
$140$	0	0
$141$	11.9150	1.00343
$142$	0	0
$143$	−3.73384	−0.312239
$144$	0	0
$145$	−4.54684	−0.377595
$146$	0	0
$147$	10.8892	0.898122
$148$	0	0
$149$	12.7887	1.04769	0.523847	−	0.851812i	$-0.324496\pi$
$149$	12.7887	1.04769	0.523847	+	0.851812i	$0.324496\pi$
$150$	0	0
$151$	−16.1354	−1.31308	−0.656542	−	0.754290i	$-0.727981\pi$
$151$	−16.1354	−1.31308	−0.656542	+	0.754290i	$0.727981\pi$
$152$	0	0
$153$	16.9998	1.37435
$154$	0	0
$155$	5.09283	0.409066
$156$	0	0
$157$	−19.6007	−1.56431	−0.782155	−	0.623084i	$-0.785879\pi$
$157$	−19.6007	−1.56431	−0.782155	+	0.623084i	$0.785879\pi$
$158$	0	0
$159$	21.6994	1.72088
$160$	0	0
$161$	6.99979	0.551661
$162$	0	0
$163$	16.1258	1.26307	0.631536	−	0.775347i	$-0.282425\pi$
$163$	16.1258	1.26307	0.631536	+	0.775347i	$0.282425\pi$
$164$	0	0
$165$	−1.80484	−0.140507
$166$	0	0
$167$	22.4615	1.73813	0.869063	−	0.494702i	$-0.164723\pi$
$167$	22.4615	1.73813	0.869063	+	0.494702i	$0.164723\pi$
$168$	0	0
$169$	18.8987	1.45375
$170$	0	0
$171$	18.2869	1.39843
$172$	0	0
$173$	4.22936	0.321552	0.160776	−	0.986991i	$-0.448600\pi$
$173$	4.22936	0.321552	0.160776	+	0.986991i	$0.448600\pi$
$174$	0	0
$175$	−8.47106	−0.640352
$176$	0	0
$177$	−48.8116	−3.66890
$178$	0	0
$179$	−18.6053	−1.39062	−0.695312	−	0.718708i	$-0.744734\pi$
$179$	−18.6053	−1.39062	−0.695312	+	0.718708i	$0.744734\pi$
$180$	0	0
$181$	−24.0845	−1.79019	−0.895095	−	0.445876i	$-0.852892\pi$
$181$	−24.0845	−1.79019	−0.895095	+	0.445876i	$0.852892\pi$
$182$	0	0
$183$	10.8346	0.800913
$184$	0	0
$185$	0.805051	0.0591885
$186$	0	0
$187$	−1.32220	−0.0966892
$188$	0	0
$189$	−36.3045	−2.64077
$190$	0	0
$191$	15.4344	1.11679	0.558397	−	0.829574i	$-0.311417\pi$
$191$	15.4344	1.11679	0.558397	+	0.829574i	$0.311417\pi$
$192$	0	0
$193$	3.68569	0.265302	0.132651	−	0.991163i	$-0.457651\pi$
$193$	3.68569	0.265302	0.132651	+	0.991163i	$0.457651\pi$
$194$	0	0
$195$	15.4190	1.10418
$196$	0	0
$197$	18.3988	1.31086	0.655432	−	0.755255i	$-0.272487\pi$
$197$	18.3988	1.31086	0.655432	+	0.755255i	$0.272487\pi$
$198$	0	0
$199$	21.1761	1.50113	0.750567	−	0.660794i	$-0.229780\pi$
$199$	21.1761	1.50113	0.750567	+	0.660794i	$0.229780\pi$
$200$	0	0
$201$	46.4342	3.27522
$202$	0	0
$203$	10.9938	0.771611
$204$	0	0
$205$	−3.98746	−0.278496
$206$	0	0
$207$	30.5660	2.12449
$208$	0	0
$209$	−1.42231	−0.0983833
$210$	0	0
$211$	8.85699	0.609740	0.304870	−	0.952394i	$-0.401387\pi$
$211$	8.85699	0.609740	0.304870	+	0.952394i	$0.401387\pi$
$212$	0	0
$213$	34.9148	2.39233
$214$	0	0
$215$	−4.05800	−0.276753
$216$	0	0
$217$	−12.3139	−0.835921
$218$	0	0
$219$	−45.7802	−3.09354
$220$	0	0
$221$	11.2958	0.759837
$222$	0	0
$223$	−3.51357	−0.235286	−0.117643	−	0.993056i	$-0.537534\pi$
$223$	−3.51357	−0.235286	−0.117643	+	0.993056i	$0.537534\pi$
$224$	0	0
$225$	−36.9906	−2.46604
$226$	0	0
$227$	18.6053	1.23488	0.617438	−	0.786620i	$-0.288171\pi$
$227$	18.6053	1.23488	0.617438	+	0.786620i	$0.288171\pi$
$228$	0	0
$229$	−9.08474	−0.600337	−0.300168	−	0.953886i	$-0.597043\pi$
$229$	−9.08474	−0.600337	−0.300168	+	0.953886i	$0.597043\pi$
$230$	0	0
$231$	4.36390	0.287124
$232$	0	0
$233$	−13.1641	−0.862409	−0.431205	−	0.902254i	$-0.641911\pi$
$233$	−13.1641	−0.862409	−0.431205	+	0.902254i	$0.641911\pi$
$234$	0	0
$235$	2.82860	0.184518
$236$	0	0
$237$	−12.1947	−0.792133
$238$	0	0
$239$	1.11085	0.0718547	0.0359273	−	0.999354i	$-0.488562\pi$
$239$	1.11085	0.0718547	0.0359273	+	0.999354i	$0.488562\pi$
$240$	0	0
$241$	2.29600	0.147899	0.0739493	−	0.997262i	$-0.476440\pi$
$241$	2.29600	0.147899	0.0739493	+	0.997262i	$0.476440\pi$
$242$	0	0
$243$	−72.0580	−4.62252
$244$	0	0
$245$	2.58506	0.165153
$246$	0	0
$247$	12.1510	0.773150
$248$	0	0
$249$	−16.3988	−1.03923
$250$	0	0
$251$	20.7663	1.31076	0.655379	−	0.755300i	$-0.272509\pi$
$251$	20.7663	1.31076	0.655379	+	0.755300i	$0.272509\pi$
$252$	0	0
$253$	−2.37736	−0.149463
$254$	0	0
$255$	5.46009	0.341924
$256$	0	0
$257$	−9.22020	−0.575140	−0.287570	−	0.957760i	$-0.592847\pi$
$257$	−9.22020	−0.575140	−0.287570	+	0.957760i	$0.592847\pi$
$258$	0	0
$259$	−1.94652	−0.120951
$260$	0	0
$261$	48.0065	2.97153
$262$	0	0
$263$	−13.5949	−0.838299	−0.419149	−	0.907917i	$-0.637672\pi$
$263$	−13.5949	−0.838299	−0.419149	+	0.907917i	$0.637672\pi$
$264$	0	0
$265$	5.15139	0.316447
$266$	0	0
$267$	−18.7063	−1.14481
$268$	0	0
$269$	−11.8735	−0.723939	−0.361970	−	0.932190i	$-0.617896\pi$
$269$	−11.8735	−0.723939	−0.361970	+	0.932190i	$0.617896\pi$
$270$	0	0
$271$	0.942732	0.0572669	0.0286334	−	0.999590i	$-0.490884\pi$
$271$	0.942732	0.0572669	0.0286334	+	0.999590i	$0.490884\pi$
$272$	0	0
$273$	−37.2815	−2.25638
$274$	0	0
$275$	2.87705	0.173492
$276$	0	0
$277$	12.3968	0.744854	0.372427	−	0.928061i	$-0.378526\pi$
$277$	12.3968	0.744854	0.372427	+	0.928061i	$0.378526\pi$
$278$	0	0
$279$	−53.7712	−3.21919
$280$	0	0
$281$	−25.0754	−1.49587	−0.747936	−	0.663771i	$-0.768955\pi$
$281$	−25.0754	−1.49587	−0.747936	+	0.663771i	$0.768955\pi$
$282$	0	0
$283$	29.9353	1.77947	0.889733	−	0.456481i	$-0.150890\pi$
$283$	29.9353	1.77947	0.889733	+	0.456481i	$0.150890\pi$
$284$	0	0
$285$	5.87348	0.347915
$286$	0	0
$287$	9.64123	0.569104
$288$	0	0
$289$	−13.0000	−0.764706
$290$	0	0
$291$	−6.78230	−0.397585
$292$	0	0
$293$	−27.2954	−1.59461	−0.797306	−	0.603575i	$-0.793742\pi$
$293$	−27.2954	−1.59461	−0.797306	+	0.603575i	$0.793742\pi$
$294$	0	0
$295$	−11.5878	−0.674665
$296$	0	0
$297$	12.3302	0.715471
$298$	0	0
$299$	20.3101	1.17456
$300$	0	0
$301$	9.81179	0.565542
$302$	0	0
$303$	−37.2716	−2.14120
$304$	0	0
$305$	2.57210	0.147278
$306$	0	0
$307$	11.9968	0.684696	0.342348	−	0.939573i	$-0.388778\pi$
$307$	11.9968	0.684696	0.342348	+	0.939573i	$0.388778\pi$
$308$	0	0
$309$	30.4203	1.73055
$310$	0	0
$311$	11.7919	0.668659	0.334330	−	0.942456i	$-0.391490\pi$
$311$	11.7919	0.668659	0.334330	+	0.942456i	$0.391490\pi$
$312$	0	0
$313$	16.4078	0.927423	0.463711	−	0.885986i	$-0.346517\pi$
$313$	16.4078	0.927423	0.463711	+	0.885986i	$0.346517\pi$
$314$	0	0
$315$	−13.3198	−0.750483
$316$	0	0
$317$	−2.90589	−0.163211	−0.0816056	−	0.996665i	$-0.526005\pi$
$317$	−2.90589	−0.163211	−0.0816056	+	0.996665i	$0.526005\pi$
$318$	0	0
$319$	−3.73384	−0.209055
$320$	0	0
$321$	−54.3025	−3.03087
$322$	0	0
$323$	4.30284	0.239417
$324$	0	0
$325$	−24.5790	−1.36340
$326$	0	0
$327$	52.1887	2.88604
$328$	0	0
$329$	−6.83925	−0.377060
$330$	0	0
$331$	14.1490	0.777698	0.388849	−	0.921302i	$-0.372873\pi$
$331$	14.1490	0.777698	0.388849	+	0.921302i	$0.372873\pi$
$332$	0	0
$333$	−8.49990	−0.465791
$334$	0	0
$335$	11.0234	0.602271
$336$	0	0
$337$	16.7201	0.910802	0.455401	−	0.890286i	$-0.349496\pi$
$337$	16.7201	0.910802	0.455401	+	0.890286i	$0.349496\pi$
$338$	0	0
$339$	−49.5443	−2.69088
$340$	0	0
$341$	4.18220	0.226479
$342$	0	0
$343$	−19.8760	−1.07321
$344$	0	0
$345$	9.81738	0.528550
$346$	0	0
$347$	−17.7857	−0.954785	−0.477393	−	0.878690i	$-0.658418\pi$
$347$	−17.7857	−0.954785	−0.477393	+	0.878690i	$0.658418\pi$
$348$	0	0
$349$	−2.51642	−0.134701	−0.0673503	−	0.997729i	$-0.521455\pi$
$349$	−2.51642	−0.134701	−0.0673503	+	0.997729i	$0.521455\pi$
$350$	0	0
$351$	−105.339	−5.62257
$352$	0	0
$353$	4.51600	0.240362	0.120181	−	0.992752i	$-0.461652\pi$
$353$	4.51600	0.240362	0.120181	+	0.992752i	$0.461652\pi$
$354$	0	0
$355$	8.28870	0.439918
$356$	0	0
$357$	−13.2019	−0.698719
$358$	0	0
$359$	−24.6801	−1.30256	−0.651282	−	0.758836i	$-0.725769\pi$
$359$	−24.6801	−1.30256	−0.651282	+	0.758836i	$0.725769\pi$
$360$	0	0
$361$	−14.3714	−0.756389
$362$	0	0
$363$	35.8205	1.88009
$364$	0	0
$365$	−10.8681	−0.568862
$366$	0	0
$367$	18.1123	0.945454	0.472727	−	0.881209i	$-0.343270\pi$
$367$	18.1123	0.945454	0.472727	+	0.881209i	$0.343270\pi$
$368$	0	0
$369$	42.1004	2.19166
$370$	0	0
$371$	−12.4555	−0.646657
$372$	0	0
$373$	15.3374	0.794138	0.397069	−	0.917789i	$-0.370027\pi$
$373$	15.3374	0.794138	0.397069	+	0.917789i	$0.370027\pi$
$374$	0	0
$375$	−25.5311	−1.31842
$376$	0	0
$377$	31.8987	1.64287
$378$	0	0
$379$	7.69399	0.395214	0.197607	−	0.980281i	$-0.436683\pi$
$379$	7.69399	0.395214	0.197607	+	0.980281i	$0.436683\pi$
$380$	0	0
$381$	−48.1167	−2.46509
$382$	0	0
$383$	14.5588	0.743918	0.371959	−	0.928249i	$-0.378686\pi$
$383$	14.5588	0.743918	0.371959	+	0.928249i	$0.378686\pi$
$384$	0	0
$385$	1.03598	0.0527984
$386$	0	0
$387$	42.8452	2.17794
$388$	0	0
$389$	8.86810	0.449630	0.224815	−	0.974401i	$-0.427822\pi$
$389$	8.86810	0.449630	0.224815	+	0.974401i	$0.427822\pi$
$390$	0	0
$391$	7.19210	0.363720
$392$	0	0
$393$	2.81200	0.141847
$394$	0	0
$395$	−2.89500	−0.145663
$396$	0	0
$397$	9.69463	0.486560	0.243280	−	0.969956i	$-0.421777\pi$
$397$	9.69463	0.486560	0.243280	+	0.969956i	$0.421777\pi$
$398$	0	0
$399$	−14.2014	−0.710961
$400$	0	0
$401$	20.8120	1.03930	0.519651	−	0.854379i	$-0.326062\pi$
$401$	20.8120	1.03930	0.519651	+	0.854379i	$0.326062\pi$
$402$	0	0
$403$	−35.7291	−1.77980
$404$	0	0
$405$	−30.3895	−1.51007
$406$	0	0
$407$	0.661102	0.0327696
$408$	0	0
$409$	12.5160	0.618876	0.309438	−	0.950920i	$-0.399859\pi$
$409$	12.5160	0.618876	0.309438	+	0.950920i	$0.399859\pi$
$410$	0	0
$411$	16.5510	0.816400
$412$	0	0
$413$	28.0179	1.37867
$414$	0	0
$415$	−3.89305	−0.191102
$416$	0	0
$417$	−3.50926	−0.171849
$418$	0	0
$419$	−2.98135	−0.145649	−0.0728243	−	0.997345i	$-0.523201\pi$
$419$	−2.98135	−0.145649	−0.0728243	+	0.997345i	$0.523201\pi$
$420$	0	0
$421$	−20.8681	−1.01705	−0.508524	−	0.861048i	$-0.669809\pi$
$421$	−20.8681	−1.01705	−0.508524	+	0.861048i	$0.669809\pi$
$422$	0	0
$423$	−29.8650	−1.45209
$424$	0	0
$425$	−8.70379	−0.422196
$426$	0	0
$427$	−6.21904	−0.300960
$428$	0	0
$429$	12.6620	0.611327
$430$	0	0
$431$	−17.2032	−0.828648	−0.414324	−	0.910129i	$-0.635982\pi$
$431$	−17.2032	−0.828648	−0.414324	+	0.910129i	$0.635982\pi$
$432$	0	0
$433$	18.2672	0.877863	0.438932	−	0.898520i	$-0.355357\pi$
$433$	18.2672	0.877863	0.438932	+	0.898520i	$0.355357\pi$
$434$	0	0
$435$	15.4190	0.739286
$436$	0	0
$437$	7.73662	0.370093
$438$	0	0
$439$	1.93761	0.0924773	0.0462387	−	0.998930i	$-0.485277\pi$
$439$	1.93761	0.0924773	0.0462387	+	0.998930i	$0.485277\pi$
$440$	0	0
$441$	−27.2936	−1.29969
$442$	0	0
$443$	−33.0662	−1.57102	−0.785511	−	0.618847i	$-0.787600\pi$
$443$	−33.0662	−1.57102	−0.785511	+	0.618847i	$0.787600\pi$
$444$	0	0
$445$	−4.44082	−0.210515
$446$	0	0
$447$	−43.3685	−2.05126
$448$	0	0
$449$	−3.57748	−0.168832	−0.0844159	−	0.996431i	$-0.526902\pi$
$449$	−3.57748	−0.168832	−0.0844159	+	0.996431i	$0.526902\pi$
$450$	0	0
$451$	−3.27447	−0.154189
$452$	0	0
$453$	54.7177	2.57086
$454$	0	0
$455$	−8.85053	−0.414919
$456$	0	0
$457$	18.9815	0.887916	0.443958	−	0.896047i	$-0.353574\pi$
$457$	18.9815	0.887916	0.443958	+	0.896047i	$0.353574\pi$
$458$	0	0
$459$	−37.3019	−1.74111
$460$	0	0
$461$	25.8301	1.20303	0.601514	−	0.798862i	$-0.294565\pi$
$461$	25.8301	1.20303	0.601514	+	0.798862i	$0.294565\pi$
$462$	0	0
$463$	2.34741	0.109094	0.0545468	−	0.998511i	$-0.482629\pi$
$463$	2.34741	0.109094	0.0545468	+	0.998511i	$0.482629\pi$
$464$	0	0
$465$	−17.2705	−0.800902
$466$	0	0
$467$	0.584373	0.0270415	0.0135208	−	0.999909i	$-0.495696\pi$
$467$	0.584373	0.0270415	0.0135208	+	0.999909i	$0.495696\pi$
$468$	0	0
$469$	−26.6533	−1.23073
$470$	0	0
$471$	66.4690	3.06273
$472$	0	0
$473$	−3.33240	−0.153224
$474$	0	0
$475$	−9.36276	−0.429593
$476$	0	0
$477$	−54.3895	−2.49032
$478$	0	0
$479$	−3.18625	−0.145583	−0.0727917	−	0.997347i	$-0.523191\pi$
$479$	−3.18625	−0.145583	−0.0727917	+	0.997347i	$0.523191\pi$
$480$	0	0
$481$	−5.64790	−0.257522
$482$	0	0
$483$	−23.7373	−1.08009
$484$	0	0
$485$	−1.61010	−0.0731109
$486$	0	0
$487$	9.84613	0.446171	0.223085	−	0.974799i	$-0.428387\pi$
$487$	9.84613	0.446171	0.223085	+	0.974799i	$0.428387\pi$
$488$	0	0
$489$	−54.6851	−2.47294
$490$	0	0
$491$	−3.26802	−0.147483	−0.0737417	−	0.997277i	$-0.523494\pi$
$491$	−3.26802	−0.147483	−0.0737417	+	0.997277i	$0.523494\pi$
$492$	0	0
$493$	11.2958	0.508737
$494$	0	0
$495$	4.52382	0.203331
$496$	0	0
$497$	−20.0411	−0.898968
$498$	0	0
$499$	25.2341	1.12963	0.564817	−	0.825216i	$-0.308947\pi$
$499$	25.2341	1.12963	0.564817	+	0.825216i	$0.308947\pi$
$500$	0	0
$501$	−76.1704	−3.40304
$502$	0	0
$503$	5.50650	0.245523	0.122761	−	0.992436i	$-0.460825\pi$
$503$	5.50650	0.245523	0.122761	+	0.992436i	$0.460825\pi$
$504$	0	0
$505$	−8.84819	−0.393739
$506$	0	0
$507$	−64.0884	−2.84627
$508$	0	0
$509$	13.2108	0.585560	0.292780	−	0.956180i	$-0.405420\pi$
$509$	13.2108	0.585560	0.292780	+	0.956180i	$0.405420\pi$
$510$	0	0
$511$	26.2778	1.16246
$512$	0	0
$513$	−40.1261	−1.77161
$514$	0	0
$515$	7.22171	0.318227
$516$	0	0
$517$	2.32283	0.102158
$518$	0	0
$519$	−14.3424	−0.629560
$520$	0	0
$521$	−10.8031	−0.473290	−0.236645	−	0.971596i	$-0.576048\pi$
$521$	−10.8031	−0.473290	−0.236645	+	0.971596i	$0.576048\pi$
$522$	0	0
$523$	5.78813	0.253097	0.126549	−	0.991960i	$-0.459610\pi$
$523$	5.78813	0.253097	0.126549	+	0.991960i	$0.459610\pi$
$524$	0	0
$525$	28.7266	1.25373
$526$	0	0
$527$	−12.6522	−0.551138
$528$	0	0
$529$	−10.0684	−0.437758
$530$	0	0
$531$	122.346	5.30936
$532$	0	0
$533$	27.9743	1.21170
$534$	0	0
$535$	−12.8913	−0.557339
$536$	0	0
$537$	63.0933	2.72268
$538$	0	0
$539$	2.12283	0.0914369
$540$	0	0
$541$	−22.8373	−0.981850	−0.490925	−	0.871202i	$-0.663341\pi$
$541$	−22.8373	−0.981850	−0.490925	+	0.871202i	$0.663341\pi$
$542$	0	0
$543$	81.6743	3.50498
$544$	0	0
$545$	12.3895	0.530707
$546$	0	0
$547$	33.0087	1.41135	0.705675	−	0.708535i	$-0.250644\pi$
$547$	33.0087	1.41135	0.705675	+	0.708535i	$0.250644\pi$
$548$	0	0
$549$	−27.1567	−1.15902
$550$	0	0
$551$	12.1510	0.517650
$552$	0	0
$553$	6.99979	0.297661
$554$	0	0
$555$	−2.73005	−0.115884
$556$	0	0
$557$	−22.5663	−0.956166	−0.478083	−	0.878315i	$-0.658668\pi$
$557$	−22.5663	−0.956166	−0.478083	+	0.878315i	$0.658668\pi$
$558$	0	0
$559$	28.4692	1.20412
$560$	0	0
$561$	4.48379	0.189306
$562$	0	0
$563$	−8.27908	−0.348922	−0.174461	−	0.984664i	$-0.555818\pi$
$563$	−8.27908	−0.348922	−0.174461	+	0.984664i	$0.555818\pi$
$564$	0	0
$565$	−11.7617	−0.494819
$566$	0	0
$567$	73.4784	3.08580
$568$	0	0
$569$	−23.6857	−0.992956	−0.496478	−	0.868049i	$-0.665374\pi$
$569$	−23.6857	−0.992956	−0.496478	+	0.868049i	$0.665374\pi$
$570$	0	0
$571$	−39.2059	−1.64072	−0.820358	−	0.571850i	$-0.806226\pi$
$571$	−39.2059	−1.64072	−0.820358	+	0.571850i	$0.806226\pi$
$572$	0	0
$573$	−52.3403	−2.18655
$574$	0	0
$575$	−15.6496	−0.652634
$576$	0	0
$577$	16.0322	0.667429	0.333715	−	0.942674i	$-0.391698\pi$
$577$	16.0322	0.667429	0.333715	+	0.942674i	$0.391698\pi$
$578$	0	0
$579$	−12.4987	−0.519430
$580$	0	0
$581$	9.41295	0.390515
$582$	0	0
$583$	4.23029	0.175201
$584$	0	0
$585$	−38.6477	−1.59789
$586$	0	0
$587$	−7.29310	−0.301019	−0.150509	−	0.988609i	$-0.548091\pi$
$587$	−7.29310	−0.301019	−0.150509	+	0.988609i	$0.548091\pi$
$588$	0	0
$589$	−13.6101	−0.560794
$590$	0	0
$591$	−62.3932	−2.56651
$592$	0	0
$593$	18.2797	0.750657	0.375328	−	0.926892i	$-0.377530\pi$
$593$	18.2797	0.750657	0.375328	+	0.926892i	$0.377530\pi$
$594$	0	0
$595$	−3.13410	−0.128486
$596$	0	0
$597$	−71.8114	−2.93904
$598$	0	0
$599$	39.0036	1.59364	0.796822	−	0.604214i	$-0.206513\pi$
$599$	39.0036	1.59364	0.796822	+	0.604214i	$0.206513\pi$
$600$	0	0
$601$	18.3030	0.746593	0.373296	−	0.927712i	$-0.378227\pi$
$601$	18.3030	0.746593	0.373296	+	0.927712i	$0.378227\pi$
$602$	0	0
$603$	−116.387	−4.73965
$604$	0	0
$605$	8.50370	0.345725
$606$	0	0
$607$	38.3464	1.55643	0.778216	−	0.627997i	$-0.216125\pi$
$607$	38.3464	1.55643	0.778216	+	0.627997i	$0.216125\pi$
$608$	0	0
$609$	−37.2815	−1.51072
$610$	0	0
$611$	−19.8443	−0.802814
$612$	0	0
$613$	18.2866	0.738590	0.369295	−	0.929312i	$-0.379599\pi$
$613$	18.2866	0.738590	0.369295	+	0.929312i	$0.379599\pi$
$614$	0	0
$615$	13.5221	0.545262
$616$	0	0
$617$	25.5629	1.02913	0.514563	−	0.857453i	$-0.327954\pi$
$617$	25.5629	1.02913	0.514563	+	0.857453i	$0.327954\pi$
$618$	0	0
$619$	−2.97560	−0.119600	−0.0597998	−	0.998210i	$-0.519046\pi$
$619$	−2.97560	−0.119600	−0.0597998	+	0.998210i	$0.519046\pi$
$620$	0	0
$621$	−67.0698	−2.69142
$622$	0	0
$623$	10.7374	0.430186
$624$	0	0
$625$	15.6984	0.627938
$626$	0	0
$627$	4.82327	0.192623
$628$	0	0
$629$	−2.00000	−0.0797452
$630$	0	0
$631$	−35.1930	−1.40101	−0.700506	−	0.713646i	$-0.747043\pi$
$631$	−35.1930	−1.40101	−0.700506	+	0.713646i	$0.747043\pi$
$632$	0	0
$633$	−30.0354	−1.19380
$634$	0	0
$635$	−11.4228	−0.453300
$636$	0	0
$637$	−18.1357	−0.718562
$638$	0	0
$639$	−87.5138	−3.46199
$640$	0	0
$641$	43.1038	1.70250	0.851249	−	0.524762i	$-0.175846\pi$
$641$	43.1038	1.70250	0.851249	+	0.524762i	$0.175846\pi$
$642$	0	0
$643$	−37.9599	−1.49699	−0.748496	−	0.663140i	$-0.769224\pi$
$643$	−37.9599	−1.49699	−0.748496	+	0.663140i	$0.769224\pi$
$644$	0	0
$645$	13.7613	0.541850
$646$	0	0
$647$	0.357647	0.0140606	0.00703028	−	0.999975i	$-0.497762\pi$
$647$	0.357647	0.0140606	0.00703028	+	0.999975i	$0.497762\pi$
$648$	0	0
$649$	−9.51579	−0.373527
$650$	0	0
$651$	41.7582	1.63663
$652$	0	0
$653$	−19.5719	−0.765907	−0.382954	−	0.923768i	$-0.625093\pi$
$653$	−19.5719	−0.765907	−0.382954	+	0.923768i	$0.625093\pi$
$654$	0	0
$655$	0.667562	0.0260838
$656$	0	0
$657$	114.748	4.47673
$658$	0	0
$659$	−35.4172	−1.37966	−0.689829	−	0.723973i	$-0.742314\pi$
$659$	−35.4172	−1.37966	−0.689829	+	0.723973i	$0.742314\pi$
$660$	0	0
$661$	0.163892	0.00637468	0.00318734	−	0.999995i	$-0.498985\pi$
$661$	0.163892	0.00637468	0.00318734	+	0.999995i	$0.498985\pi$
$662$	0	0
$663$	−38.3057	−1.48767
$664$	0	0
$665$	−3.37138	−0.130737
$666$	0	0
$667$	20.3101	0.786411
$668$	0	0
$669$	11.9150	0.460662
$670$	0	0
$671$	2.11219	0.0815401
$672$	0	0
$673$	−19.1981	−0.740033	−0.370017	−	0.929025i	$-0.620648\pi$
$673$	−19.1981	−0.740033	−0.370017	+	0.929025i	$0.620648\pi$
$674$	0	0
$675$	81.1670	3.12412
$676$	0	0
$677$	14.7704	0.567674	0.283837	−	0.958873i	$-0.408393\pi$
$677$	14.7704	0.567674	0.283837	+	0.958873i	$0.408393\pi$
$678$	0	0
$679$	3.89305	0.149401
$680$	0	0
$681$	−63.0933	−2.41774
$682$	0	0
$683$	48.3053	1.84835	0.924176	−	0.381968i	$-0.124753\pi$
$683$	48.3053	1.84835	0.924176	+	0.381968i	$0.124753\pi$
$684$	0	0
$685$	3.92916	0.150126
$686$	0	0
$687$	30.8077	1.17539
$688$	0	0
$689$	−36.1400	−1.37682
$690$	0	0
$691$	24.8147	0.943995	0.471998	−	0.881600i	$-0.343533\pi$
$691$	24.8147	0.943995	0.471998	+	0.881600i	$0.343533\pi$
$692$	0	0
$693$	−10.9381	−0.415504
$694$	0	0
$695$	−0.833089	−0.0316009
$696$	0	0
$697$	9.90610	0.375220
$698$	0	0
$699$	44.6414	1.68849
$700$	0	0
$701$	46.4143	1.75304	0.876522	−	0.481361i	$-0.159858\pi$
$701$	46.4143	1.75304	0.876522	+	0.481361i	$0.159858\pi$
$702$	0	0
$703$	−2.15142	−0.0811424
$704$	0	0
$705$	−9.59222	−0.361264
$706$	0	0
$707$	21.3939	0.804602
$708$	0	0
$709$	−46.2265	−1.73607	−0.868037	−	0.496500i	$-0.834618\pi$
$709$	−46.2265	−1.73607	−0.868037	+	0.496500i	$0.834618\pi$
$710$	0	0
$711$	30.5660	1.14632
$712$	0	0
$713$	−22.7490	−0.851955
$714$	0	0
$715$	3.00593	0.112415
$716$	0	0
$717$	−3.76705	−0.140683
$718$	0	0
$719$	32.7846	1.22266	0.611329	−	0.791376i	$-0.290635\pi$
$719$	32.7846	1.22266	0.611329	+	0.791376i	$0.290635\pi$
$720$	0	0
$721$	−17.4613	−0.650292
$722$	0	0
$723$	−7.78609	−0.289568
$724$	0	0
$725$	−24.5790	−0.912843
$726$	0	0
$727$	36.1796	1.34183	0.670913	−	0.741536i	$-0.265902\pi$
$727$	36.1796	1.34183	0.670913	+	0.741536i	$0.265902\pi$
$728$	0	0
$729$	131.114	4.85607
$730$	0	0
$731$	10.0814	0.372872
$732$	0	0
$733$	−4.56833	−0.168735	−0.0843675	−	0.996435i	$-0.526887\pi$
$733$	−4.56833	−0.168735	−0.0843675	+	0.996435i	$0.526887\pi$
$734$	0	0
$735$	−8.76632	−0.323351
$736$	0	0
$737$	9.05232	0.333447
$738$	0	0
$739$	−54.2827	−1.99682	−0.998410	−	0.0563674i	$-0.982048\pi$
$739$	−54.2827	−1.99682	−0.998410	+	0.0563674i	$0.982048\pi$
$740$	0	0
$741$	−41.2059	−1.51374
$742$	0	0
$743$	−18.4932	−0.678449	−0.339225	−	0.940705i	$-0.610165\pi$
$743$	−18.4932	−0.678449	−0.339225	+	0.940705i	$0.610165\pi$
$744$	0	0
$745$	−10.2956	−0.377201
$746$	0	0
$747$	41.1036	1.50390
$748$	0	0
$749$	31.1697	1.13892
$750$	0	0
$751$	24.2703	0.885635	0.442817	−	0.896612i	$-0.353979\pi$
$751$	24.2703	0.885635	0.442817	+	0.896612i	$0.353979\pi$
$752$	0	0
$753$	−70.4217	−2.56631
$754$	0	0
$755$	12.9898	0.472749
$756$	0	0
$757$	−12.0065	−0.436385	−0.218192	−	0.975906i	$-0.570016\pi$
$757$	−12.0065	−0.436385	−0.218192	+	0.975906i	$0.570016\pi$
$758$	0	0
$759$	8.06197	0.292631
$760$	0	0
$761$	7.63895	0.276912	0.138456	−	0.990369i	$-0.455786\pi$
$761$	7.63895	0.276912	0.138456	+	0.990369i	$0.455786\pi$
$762$	0	0
$763$	−29.9564	−1.08449
$764$	0	0
$765$	−13.6857	−0.494807
$766$	0	0
$767$	81.2948	2.93538
$768$	0	0
$769$	22.7611	0.820785	0.410393	−	0.911909i	$-0.365392\pi$
$769$	22.7611	0.820785	0.410393	+	0.911909i	$0.365392\pi$
$770$	0	0
$771$	31.2671	1.12606
$772$	0	0
$773$	50.0841	1.80140	0.900700	−	0.434441i	$-0.143054\pi$
$773$	50.0841	1.80140	0.900700	+	0.434441i	$0.143054\pi$
$774$	0	0
$775$	27.5305	0.988925
$776$	0	0
$777$	6.60095	0.236808
$778$	0	0
$779$	10.6561	0.381795
$780$	0	0
$781$	6.80662	0.243560
$782$	0	0
$783$	−105.339	−3.76450
$784$	0	0
$785$	15.7796	0.563198
$786$	0	0
$787$	−8.15264	−0.290610	−0.145305	−	0.989387i	$-0.546416\pi$
$787$	−8.15264	−0.290610	−0.145305	+	0.989387i	$0.546416\pi$
$788$	0	0
$789$	46.1024	1.64129
$790$	0	0
$791$	28.4385	1.01116
$792$	0	0
$793$	−18.0447	−0.640788
$794$	0	0
$795$	−17.4691	−0.619566
$796$	0	0
$797$	23.8930	0.846333	0.423166	−	0.906052i	$-0.360919\pi$
$797$	23.8930	0.846333	0.423166	+	0.906052i	$0.360919\pi$
$798$	0	0
$799$	−7.02714	−0.248602
$800$	0	0
$801$	46.8872	1.65668
$802$	0	0
$803$	−8.92481	−0.314950
$804$	0	0
$805$	−5.63519	−0.198614
$806$	0	0
$807$	40.2648	1.41739
$808$	0	0
$809$	24.8299	0.872972	0.436486	−	0.899711i	$-0.356223\pi$
$809$	24.8299	0.872972	0.436486	+	0.899711i	$0.356223\pi$
$810$	0	0
$811$	−10.7425	−0.377219	−0.188609	−	0.982052i	$-0.560398\pi$
$811$	−10.7425	−0.377219	−0.188609	+	0.982052i	$0.560398\pi$
$812$	0	0
$813$	−3.19694	−0.112122
$814$	0	0
$815$	−12.9821	−0.454743
$816$	0	0
$817$	10.8446	0.379405
$818$	0	0
$819$	93.4458	3.26526
$820$	0	0
$821$	41.7881	1.45842	0.729208	−	0.684293i	$-0.239889\pi$
$821$	41.7881	1.45842	0.729208	+	0.684293i	$0.239889\pi$
$822$	0	0
$823$	−12.7258	−0.443592	−0.221796	−	0.975093i	$-0.571192\pi$
$823$	−12.7258	−0.443592	−0.221796	+	0.975093i	$0.571192\pi$
$824$	0	0
$825$	−9.75650	−0.339678
$826$	0	0
$827$	−1.99796	−0.0694760	−0.0347380	−	0.999396i	$-0.511060\pi$
$827$	−1.99796	−0.0694760	−0.0347380	+	0.999396i	$0.511060\pi$
$828$	0	0
$829$	−39.0010	−1.35456	−0.677280	−	0.735725i	$-0.736841\pi$
$829$	−39.0010	−1.35456	−0.677280	+	0.735725i	$0.736841\pi$
$830$	0	0
$831$	−42.0396	−1.45834
$832$	0	0
$833$	−6.42210	−0.222513
$834$	0	0
$835$	−18.0827	−0.625777
$836$	0	0
$837$	117.988	4.07826
$838$	0	0
$839$	4.39565	0.151755	0.0758774	−	0.997117i	$-0.475824\pi$
$839$	4.39565	0.151755	0.0758774	+	0.997117i	$0.475824\pi$
$840$	0	0
$841$	2.89874	0.0999564
$842$	0	0
$843$	85.0344	2.92874
$844$	0	0
$845$	−15.2144	−0.523393
$846$	0	0
$847$	−20.5610	−0.706485
$848$	0	0
$849$	−101.515	−3.48398
$850$	0	0
$851$	−3.59605	−0.123271
$852$	0	0
$853$	45.1945	1.54743	0.773716	−	0.633533i	$-0.218396\pi$
$853$	45.1945	1.54743	0.773716	+	0.633533i	$0.218396\pi$
$854$	0	0
$855$	−14.7218	−0.503477
$856$	0	0
$857$	44.1874	1.50941	0.754706	−	0.656063i	$-0.227780\pi$
$857$	44.1874	1.50941	0.754706	+	0.656063i	$0.227780\pi$
$858$	0	0
$859$	−17.9506	−0.612468	−0.306234	−	0.951956i	$-0.599069\pi$
$859$	−17.9506	−0.612468	−0.306234	+	0.951956i	$0.599069\pi$
$860$	0	0
$861$	−32.6948	−1.11424
$862$	0	0
$863$	−5.12244	−0.174370	−0.0871850	−	0.996192i	$-0.527787\pi$
$863$	−5.12244	−0.174370	−0.0871850	+	0.996192i	$0.527787\pi$
$864$	0	0
$865$	−3.40484	−0.115768
$866$	0	0
$867$	44.0849	1.49720
$868$	0	0
$869$	−2.37736	−0.0806463
$870$	0	0
$871$	−77.3353	−2.62041
$872$	0	0
$873$	16.9998	0.575356
$874$	0	0
$875$	14.6549	0.495425
$876$	0	0
$877$	−55.3710	−1.86974	−0.934872	−	0.354984i	$-0.884486\pi$
$877$	−55.3710	−1.86974	−0.934872	+	0.354984i	$0.884486\pi$
$878$	0	0
$879$	92.5627	3.12206
$880$	0	0
$881$	15.9131	0.536124	0.268062	−	0.963402i	$-0.413617\pi$
$881$	15.9131	0.536124	0.268062	+	0.963402i	$0.413617\pi$
$882$	0	0
$883$	15.8074	0.531962	0.265981	−	0.963978i	$-0.414304\pi$
$883$	15.8074	0.531962	0.265981	+	0.963978i	$0.414304\pi$
$884$	0	0
$885$	39.2958	1.32091
$886$	0	0
$887$	−42.5475	−1.42861	−0.714303	−	0.699837i	$-0.753256\pi$
$887$	−42.5475	−1.42861	−0.714303	+	0.699837i	$0.753256\pi$
$888$	0	0
$889$	27.6190	0.926313
$890$	0	0
$891$	−24.9556	−0.836046
$892$	0	0
$893$	−7.55918	−0.252958
$894$	0	0
$895$	14.9782	0.500666
$896$	0	0
$897$	−68.8746	−2.29966
$898$	0	0
$899$	−35.7291	−1.19163
$900$	0	0
$901$	−12.7977	−0.426353
$902$	0	0
$903$	−33.2732	−1.10726
$904$	0	0
$905$	19.3893	0.644521
$906$	0	0
$907$	20.4287	0.678323	0.339161	−	0.940728i	$-0.389857\pi$
$907$	20.4287	0.678323	0.339161	+	0.940728i	$0.389857\pi$
$908$	0	0
$909$	93.4211	3.09858
$910$	0	0
$911$	−52.2590	−1.73142	−0.865709	−	0.500548i	$-0.833132\pi$
$911$	−52.2590	−1.73142	−0.865709	+	0.500548i	$0.833132\pi$
$912$	0	0
$913$	−3.19694	−0.105803
$914$	0	0
$915$	−8.72236	−0.288352
$916$	0	0
$917$	−1.61409	−0.0533019
$918$	0	0
$919$	35.4997	1.17103	0.585513	−	0.810663i	$-0.300893\pi$
$919$	35.4997	1.17103	0.585513	+	0.810663i	$0.300893\pi$
$920$	0	0
$921$	−40.6831	−1.34055
$922$	0	0
$923$	−58.1500	−1.91403
$924$	0	0
$925$	4.35189	0.143089
$926$	0	0
$927$	−76.2484	−2.50432
$928$	0	0
$929$	−17.0823	−0.560453	−0.280227	−	0.959934i	$-0.590410\pi$
$929$	−17.0823	−0.560453	−0.280227	+	0.959934i	$0.590410\pi$
$930$	0	0
$931$	−6.90832	−0.226411
$932$	0	0
$933$	−39.9882	−1.30916
$934$	0	0
$935$	1.06444	0.0348110
$936$	0	0
$937$	−8.24885	−0.269478	−0.134739	−	0.990881i	$-0.543020\pi$
$937$	−8.24885	−0.269478	−0.134739	+	0.990881i	$0.543020\pi$
$938$	0	0
$939$	−55.6412	−1.81578
$940$	0	0
$941$	10.1122	0.329648	0.164824	−	0.986323i	$-0.447294\pi$
$941$	10.1122	0.329648	0.164824	+	0.986323i	$0.447294\pi$
$942$	0	0
$943$	17.8114	0.580020
$944$	0	0
$945$	29.2270	0.950754
$946$	0	0
$947$	12.9003	0.419205	0.209602	−	0.977787i	$-0.432783\pi$
$947$	12.9003	0.419205	0.209602	+	0.977787i	$0.432783\pi$
$948$	0	0
$949$	76.2460	2.47505
$950$	0	0
$951$	9.85432	0.319548
$952$	0	0
$953$	41.1422	1.33273	0.666363	−	0.745627i	$-0.267850\pi$
$953$	41.1422	1.33273	0.666363	+	0.745627i	$0.267850\pi$
$954$	0	0
$955$	−12.4255	−0.402079
$956$	0	0
$957$	12.6620	0.409304
$958$	0	0
$959$	−9.50028	−0.306780
$960$	0	0
$961$	9.01949	0.290951
$962$	0	0
$963$	136.109	4.38605
$964$	0	0
$965$	−2.96717	−0.0955165
$966$	0	0
$967$	47.7659	1.53605	0.768024	−	0.640421i	$-0.221240\pi$
$967$	47.7659	1.53605	0.768024	+	0.640421i	$0.221240\pi$
$968$	0	0
$969$	−14.5916	−0.468749
$970$	0	0
$971$	−26.3450	−0.845450	−0.422725	−	0.906258i	$-0.638926\pi$
$971$	−26.3450	−0.845450	−0.422725	+	0.906258i	$0.638926\pi$
$972$	0	0
$973$	2.01432	0.0645760
$974$	0	0
$975$	83.3512	2.66938
$976$	0	0
$977$	16.8625	0.539479	0.269740	−	0.962933i	$-0.413062\pi$
$977$	16.8625	0.539479	0.269740	+	0.962933i	$0.413062\pi$
$978$	0	0
$979$	−3.64678	−0.116551
$980$	0	0
$981$	−130.811	−4.17647
$982$	0	0
$983$	−32.8038	−1.04628	−0.523140	−	0.852247i	$-0.675239\pi$
$983$	−32.8038	−1.04628	−0.523140	+	0.852247i	$0.675239\pi$
$984$	0	0
$985$	−14.8120	−0.471950
$986$	0	0
$987$	23.1929	0.738238
$988$	0	0
$989$	18.1265	0.576390
$990$	0	0
$991$	5.89131	0.187144	0.0935718	−	0.995613i	$-0.470172\pi$
$991$	5.89131	0.187144	0.0935718	+	0.995613i	$0.470172\pi$
$992$	0	0
$993$	−47.9813	−1.52264
$994$	0	0
$995$	−17.0478	−0.540453
$996$	0	0
$997$	−33.2015	−1.05150	−0.525751	−	0.850639i	$-0.676216\pi$
$997$	−33.2015	−1.05150	−0.525751	+	0.850639i	$0.676216\pi$
$998$	0	0
$999$	18.6510	0.590090

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2368.2.a.bj.1.1		8
4.3	odd	2	inner	2368.2.a.bj.1.8		8
8.3	odd	2		1184.2.a.p.1.1	✓	8
8.5	even	2		1184.2.a.p.1.8	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1184.2.a.p.1.1	✓	8	8.3	odd	2
1184.2.a.p.1.8	yes	8	8.5	even	2
2368.2.a.bj.1.1		8	1.1	even	1	trivial
2368.2.a.bj.1.8		8	4.3	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2368 = 2^{6} \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2368.a (trivial)

\(f(q)\)	\(=\)	\(q-3.39115 q^{3} -0.805051 q^{5} +1.94652 q^{7} +8.49990 q^{9} +O(q^{10})\)	\(q-3.39115 q^{3} -0.805051 q^{5} +1.94652 q^{7} +8.49990 q^{9} -0.661102 q^{11} +5.64790 q^{13} +2.73005 q^{15} +2.00000 q^{17} +2.15142 q^{19} -6.60095 q^{21} +3.59605 q^{23} -4.35189 q^{25} -18.6510 q^{27} +5.64790 q^{29} -6.32610 q^{31} +2.24190 q^{33} -1.56705 q^{35} -1.00000 q^{37} -19.1529 q^{39} +4.95305 q^{41} +5.04068 q^{43} -6.84285 q^{45} -3.51357 q^{47} -3.21105 q^{49} -6.78230 q^{51} -6.39884 q^{53} +0.532221 q^{55} -7.29579 q^{57} +14.3938 q^{59} -3.19495 q^{61} +16.5452 q^{63} -4.54684 q^{65} -13.6928 q^{67} -12.1947 q^{69} -10.2959 q^{71} +13.4999 q^{73} +14.7579 q^{75} -1.28685 q^{77} +3.59605 q^{79} +37.7485 q^{81} +4.83578 q^{83} -1.61010 q^{85} -19.1529 q^{87} +5.51621 q^{89} +10.9938 q^{91} +21.4527 q^{93} -1.73200 q^{95} +2.00000 q^{97} -5.61930 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 2 q^{5} + 16 q^{9}+O(q^{10}) \) 8 * q + 2 * q^5 + 16 * q^9	\( 8 q + 2 q^{5} + 16 q^{9} - 2 q^{13} + 16 q^{17} - 2 q^{21} + 22 q^{25} - 2 q^{29} + 30 q^{33} - 8 q^{37} + 36 q^{41} - 16 q^{45} + 42 q^{49} + 2 q^{53} + 36 q^{57} - 34 q^{61} + 12 q^{65} - 2 q^{69} + 56 q^{73} + 14 q^{77} + 48 q^{81} + 4 q^{85} + 20 q^{89} + 12 q^{93} + 16 q^{97}+O(q^{100}) \) 8 * q + 2 * q^5 + 16 * q^9 - 2 * q^13 + 16 * q^17 - 2 * q^21 + 22 * q^25 - 2 * q^29 + 30 * q^33 - 8 * q^37 + 36 * q^41 - 16 * q^45 + 42 * q^49 + 2 * q^53 + 36 * q^57 - 34 * q^61 + 12 * q^65 - 2 * q^69 + 56 * q^73 + 14 * q^77 + 48 * q^81 + 4 * q^85 + 20 * q^89 + 12 * q^93 + 16 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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