LMFDB - Embedded newform 1856.4.a.bh.1.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1856,4,Mod(1,1856)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1856, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("1856.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1856 = 2^{6} \cdot 29 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1856.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:	$109.507544971$
Analytic rank:	$1$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 164x^{8} + 8446x^{6} - 181180x^{4} + 1701497x^{2} - 5718568 $ x^10 - 164x^8 + 8446x^6 - 181180x^4 + 1701497x^2 - 5718568
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{11} $
Twist minimal:	no (minimal twist has level 928)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$3.81023$ of defining polynomial
Character	$\chi$	$=$	1856.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.81023 q^{3} -15.9115 q^{5} -24.5603 q^{7} -12.4821 q^{9} +O(q^{10})$	$q+3.81023 q^{3} -15.9115 q^{5} -24.5603 q^{7} -12.4821 q^{9} +12.0272 q^{11} +50.0577 q^{13} -60.6264 q^{15} +40.0316 q^{17} +138.257 q^{19} -93.5806 q^{21} +68.4153 q^{23} +128.174 q^{25} -150.436 q^{27} -29.0000 q^{29} +26.3636 q^{31} +45.8266 q^{33} +390.791 q^{35} -337.253 q^{37} +190.732 q^{39} +459.353 q^{41} -64.3753 q^{43} +198.609 q^{45} -492.487 q^{47} +260.210 q^{49} +152.530 q^{51} +319.834 q^{53} -191.371 q^{55} +526.790 q^{57} -284.951 q^{59} +326.831 q^{61} +306.565 q^{63} -796.492 q^{65} -326.036 q^{67} +260.678 q^{69} +94.1730 q^{71} +134.234 q^{73} +488.375 q^{75} -295.393 q^{77} +641.476 q^{79} -236.179 q^{81} -850.002 q^{83} -636.960 q^{85} -110.497 q^{87} -93.6408 q^{89} -1229.43 q^{91} +100.452 q^{93} -2199.87 q^{95} +740.936 q^{97} -150.125 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 40 q^{5} + 58 q^{9}+O(q^{10}) $ 10 * q - 40 * q^5 + 58 * q^9	$ 10 q - 40 q^{5} + 58 q^{9} - 136 q^{13} + 144 q^{17} - 92 q^{21} + 334 q^{25} - 290 q^{29} + 536 q^{33} - 212 q^{37} + 400 q^{41} - 748 q^{45} + 626 q^{49} - 1304 q^{53} + 408 q^{57} - 1600 q^{61} + 1808 q^{65} - 1916 q^{69} + 1332 q^{73} - 1724 q^{77} + 730 q^{81} - 3484 q^{85} + 2256 q^{89} - 3500 q^{93} + 560 q^{97}+O(q^{100}) $ 10 * q - 40 * q^5 + 58 * q^9 - 136 * q^13 + 144 * q^17 - 92 * q^21 + 334 * q^25 - 290 * q^29 + 536 * q^33 - 212 * q^37 + 400 * q^41 - 748 * q^45 + 626 * q^49 - 1304 * q^53 + 408 * q^57 - 1600 * q^61 + 1808 * q^65 - 1916 * q^69 + 1332 * q^73 - 1724 * q^77 + 730 * q^81 - 3484 * q^85 + 2256 * q^89 - 3500 * q^93 + 560 * 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
$3$	3.81023	0.733280	0.366640	−	0.930363i	$-0.380508\pi$
$3$	3.81023	0.733280	0.366640	+	0.930363i	$0.380508\pi$
$4$	0	0
$5$	−15.9115	−1.42316	−0.711582	−	0.702603i	$-0.752021\pi$
$5$	−15.9115	−1.42316	−0.711582	+	0.702603i	$0.752021\pi$
$6$	0	0
$7$	−24.5603	−1.32613	−0.663067	−	0.748560i	$-0.730745\pi$
$7$	−24.5603	−1.32613	−0.663067	+	0.748560i	$0.730745\pi$
$8$	0	0
$9$	−12.4821	−0.462301
$10$	0	0
$11$	12.0272	0.329668	0.164834	−	0.986321i	$-0.447291\pi$
$11$	12.0272	0.329668	0.164834	+	0.986321i	$0.447291\pi$
$12$	0	0
$13$	50.0577	1.06796	0.533982	−	0.845496i	$-0.320695\pi$
$13$	50.0577	1.06796	0.533982	+	0.845496i	$0.320695\pi$
$14$	0	0
$15$	−60.6264	−1.04358
$16$	0	0
$17$	40.0316	0.571122	0.285561	−	0.958361i	$-0.407820\pi$
$17$	40.0316	0.571122	0.285561	+	0.958361i	$0.407820\pi$
$18$	0	0
$19$	138.257	1.66938	0.834691	−	0.550719i	$-0.185646\pi$
$19$	138.257	1.66938	0.834691	+	0.550719i	$0.185646\pi$
$20$	0	0
$21$	−93.5806	−0.972427
$22$	0	0
$23$	68.4153	0.620243	0.310121	−	0.950697i	$-0.399630\pi$
$23$	68.4153	0.620243	0.310121	+	0.950697i	$0.399630\pi$
$24$	0	0
$25$	128.174	1.02540
$26$	0	0
$27$	−150.436	−1.07228
$28$	0	0
$29$	−29.0000	−0.185695
$29$	−29.0000	−0.185695
$30$	0	0
$31$	26.3636	0.152744	0.0763718	−	0.997079i	$-0.475666\pi$
$31$	26.3636	0.152744	0.0763718	+	0.997079i	$0.475666\pi$
$32$	0	0
$33$	45.8266	0.241739
$34$	0	0
$35$	390.791	1.88730
$36$	0	0
$37$	−337.253	−1.49849	−0.749243	−	0.662295i	$-0.769582\pi$
$37$	−337.253	−1.49849	−0.749243	+	0.662295i	$0.769582\pi$
$38$	0	0
$39$	190.732	0.783116
$40$	0	0
$41$	459.353	1.74973	0.874864	−	0.484368i	$-0.160950\pi$
$41$	459.353	1.74973	0.874864	+	0.484368i	$0.160950\pi$
$42$	0	0
$43$	−64.3753	−0.228306	−0.114153	−	0.993463i	$-0.536415\pi$
$43$	−64.3753	−0.228306	−0.114153	+	0.993463i	$0.536415\pi$
$44$	0	0
$45$	198.609	0.657930
$46$	0	0
$47$	−492.487	−1.52844	−0.764219	−	0.644957i	$-0.776875\pi$
$47$	−492.487	−1.52844	−0.764219	+	0.644957i	$0.776875\pi$
$48$	0	0
$49$	260.210	0.758629
$50$	0	0
$51$	152.530	0.418792
$52$	0	0
$53$	319.834	0.828917	0.414459	−	0.910068i	$-0.363971\pi$
$53$	319.834	0.828917	0.414459	+	0.910068i	$0.363971\pi$
$54$	0	0
$55$	−191.371	−0.469172
$56$	0	0
$57$	526.790	1.22412
$58$	0	0
$59$	−284.951	−0.628770	−0.314385	−	0.949296i	$-0.601798\pi$
$59$	−284.951	−0.628770	−0.314385	+	0.949296i	$0.601798\pi$
$60$	0	0
$61$	326.831	0.686006	0.343003	−	0.939334i	$-0.388556\pi$
$61$	326.831	0.686006	0.343003	+	0.939334i	$0.388556\pi$
$62$	0	0
$63$	306.565	0.613072
$64$	0	0
$65$	−796.492	−1.51989
$66$	0	0
$67$	−326.036	−0.594502	−0.297251	−	0.954799i	$-0.596070\pi$
$67$	−326.036	−0.594502	−0.297251	+	0.954799i	$0.596070\pi$
$68$	0	0
$69$	260.678	0.454811
$70$	0	0
$71$	94.1730	0.157412	0.0787062	−	0.996898i	$-0.474921\pi$
$71$	94.1730	0.157412	0.0787062	+	0.996898i	$0.474921\pi$
$72$	0	0
$73$	134.234	0.215219	0.107609	−	0.994193i	$-0.465680\pi$
$73$	134.234	0.215219	0.107609	+	0.994193i	$0.465680\pi$
$74$	0	0
$75$	488.375	0.751902
$76$	0	0
$77$	−295.393	−0.437184
$78$	0	0
$79$	641.476	0.913566	0.456783	−	0.889578i	$-0.349002\pi$
$79$	641.476	0.913566	0.456783	+	0.889578i	$0.349002\pi$
$80$	0	0
$81$	−236.179	−0.323977
$82$	0	0
$83$	−850.002	−1.12409	−0.562047	−	0.827105i	$-0.689986\pi$
$83$	−850.002	−1.12409	−0.562047	+	0.827105i	$0.689986\pi$
$84$	0	0
$85$	−636.960	−0.812801
$86$	0	0
$87$	−110.497	−0.136167
$88$	0	0
$89$	−93.6408	−0.111527	−0.0557635	−	0.998444i	$-0.517759\pi$
$89$	−93.6408	−0.111527	−0.0557635	+	0.998444i	$0.517759\pi$
$90$	0	0
$91$	−1229.43	−1.41626
$92$	0	0
$93$	100.452	0.112004
$94$	0	0
$95$	−2199.87	−2.37580
$96$	0	0
$97$	740.936	0.775574	0.387787	−	0.921749i	$-0.373240\pi$
$97$	740.936	0.775574	0.387787	+	0.921749i	$0.373240\pi$
$98$	0	0
$99$	−150.125	−0.152406
$100$	0	0
$101$	−458.758	−0.451962	−0.225981	−	0.974132i	$-0.572559\pi$
$101$	−458.758	−0.451962	−0.225981	+	0.974132i	$0.572559\pi$
$102$	0	0
$103$	−522.574	−0.499910	−0.249955	−	0.968257i	$-0.580416\pi$
$103$	−522.574	−0.499910	−0.249955	+	0.968257i	$0.580416\pi$
$104$	0	0
$105$	1489.00	1.38392
$106$	0	0
$107$	1371.61	1.23924	0.619620	−	0.784902i	$-0.287287\pi$
$107$	1371.61	1.23924	0.619620	+	0.784902i	$0.287287\pi$
$108$	0	0
$109$	−1603.69	−1.40923	−0.704614	−	0.709591i	$-0.748880\pi$
$109$	−1603.69	−1.40923	−0.704614	+	0.709591i	$0.748880\pi$
$110$	0	0
$111$	−1285.01	−1.09881
$112$	0	0
$113$	−1965.60	−1.63635	−0.818177	−	0.574967i	$-0.805015\pi$
$113$	−1965.60	−1.63635	−0.818177	+	0.574967i	$0.805015\pi$
$114$	0	0
$115$	−1088.59	−0.882707
$116$	0	0
$117$	−624.827	−0.493720
$118$	0	0
$119$	−983.188	−0.757384
$120$	0	0
$121$	−1186.35	−0.891319
$122$	0	0
$123$	1750.24	1.28304
$124$	0	0
$125$	−50.5099	−0.0361420
$126$	0	0
$127$	1976.73	1.38115	0.690577	−	0.723259i	$-0.257357\pi$
$127$	1976.73	1.38115	0.690577	+	0.723259i	$0.257357\pi$
$128$	0	0
$129$	−245.285	−0.167412
$130$	0	0
$131$	−2141.42	−1.42822	−0.714110	−	0.700033i	$-0.753168\pi$
$131$	−2141.42	−1.42822	−0.714110	+	0.700033i	$0.753168\pi$
$132$	0	0
$133$	−3395.63	−2.21382
$134$	0	0
$135$	2393.66	1.52602
$136$	0	0
$137$	1927.94	1.20230	0.601151	−	0.799136i	$-0.294709\pi$
$137$	1927.94	1.20230	0.601151	+	0.799136i	$0.294709\pi$
$138$	0	0
$139$	−1650.79	−1.00733	−0.503663	−	0.863900i	$-0.668015\pi$
$139$	−1650.79	−1.00733	−0.503663	+	0.863900i	$0.668015\pi$
$140$	0	0
$141$	−1876.49	−1.12077
$142$	0	0
$143$	602.057	0.352073
$144$	0	0
$145$	461.432	0.264275
$146$	0	0
$147$	991.460	0.556287
$148$	0	0
$149$	−1951.44	−1.07294	−0.536470	−	0.843919i	$-0.680243\pi$
$149$	−1951.44	−1.07294	−0.536470	+	0.843919i	$0.680243\pi$
$150$	0	0
$151$	−2418.87	−1.30361	−0.651803	−	0.758388i	$-0.725987\pi$
$151$	−2418.87	−1.30361	−0.651803	+	0.758388i	$0.725987\pi$
$152$	0	0
$153$	−499.679	−0.264030
$154$	0	0
$155$	−419.484	−0.217379
$156$	0	0
$157$	−3306.41	−1.68077	−0.840383	−	0.541993i	$-0.817670\pi$
$157$	−3306.41	−1.68077	−0.840383	+	0.541993i	$0.817670\pi$
$158$	0	0
$159$	1218.64	0.607828
$160$	0	0
$161$	−1680.30	−0.822524
$162$	0	0
$163$	2005.44	0.963670	0.481835	−	0.876262i	$-0.339970\pi$
$163$	2005.44	0.963670	0.481835	+	0.876262i	$0.339970\pi$
$164$	0	0
$165$	−729.168	−0.344034
$166$	0	0
$167$	2348.01	1.08799	0.543995	−	0.839089i	$-0.316911\pi$
$167$	2348.01	1.08799	0.543995	+	0.839089i	$0.316911\pi$
$168$	0	0
$169$	308.778	0.140545
$170$	0	0
$171$	−1725.74	−0.771757
$172$	0	0
$173$	−56.4715	−0.0248176	−0.0124088	−	0.999923i	$-0.503950\pi$
$173$	−56.4715	−0.0248176	−0.0124088	+	0.999923i	$0.503950\pi$
$174$	0	0
$175$	−3148.01	−1.35981
$176$	0	0
$177$	−1085.73	−0.461064
$178$	0	0
$179$	−2618.63	−1.09344	−0.546720	−	0.837315i	$-0.684124\pi$
$179$	−2618.63	−1.09344	−0.546720	+	0.837315i	$0.684124\pi$
$180$	0	0
$181$	−3065.16	−1.25874	−0.629370	−	0.777106i	$-0.716687\pi$
$181$	−3065.16	−1.25874	−0.629370	+	0.777106i	$0.716687\pi$
$182$	0	0
$183$	1245.30	0.503034
$184$	0	0
$185$	5366.18	2.13259
$186$	0	0
$187$	481.469	0.188281
$188$	0	0
$189$	3694.76	1.42198
$190$	0	0
$191$	−2943.25	−1.11501	−0.557503	−	0.830175i	$-0.688240\pi$
$191$	−2943.25	−1.11501	−0.557503	+	0.830175i	$0.688240\pi$
$192$	0	0
$193$	142.841	0.0532742	0.0266371	−	0.999645i	$-0.491520\pi$
$193$	142.841	0.0532742	0.0266371	+	0.999645i	$0.491520\pi$
$194$	0	0
$195$	−3034.82	−1.11450
$196$	0	0
$197$	−736.572	−0.266389	−0.133194	−	0.991090i	$-0.542523\pi$
$197$	−736.572	−0.266389	−0.133194	+	0.991090i	$0.542523\pi$
$198$	0	0
$199$	3301.43	1.17604	0.588021	−	0.808846i	$-0.299907\pi$
$199$	3301.43	1.17604	0.588021	+	0.808846i	$0.299907\pi$
$200$	0	0
$201$	−1242.27	−0.435937
$202$	0	0
$203$	712.250	0.246257
$204$	0	0
$205$	−7308.97	−2.49015
$206$	0	0
$207$	−853.968	−0.286739
$208$	0	0
$209$	1662.85	0.550342
$210$	0	0
$211$	−3333.97	−1.08777	−0.543887	−	0.839159i	$-0.683048\pi$
$211$	−3333.97	−1.08777	−0.543887	+	0.839159i	$0.683048\pi$
$212$	0	0
$213$	358.821	0.115427
$214$	0	0
$215$	1024.30	0.324916
$216$	0	0
$217$	−647.500	−0.202558
$218$	0	0
$219$	511.465	0.157815
$220$	0	0
$221$	2003.89	0.609937
$222$	0	0
$223$	−1569.24	−0.471230	−0.235615	−	0.971846i	$-0.575710\pi$
$223$	−1569.24	−0.471230	−0.235615	+	0.971846i	$0.575710\pi$
$224$	0	0
$225$	−1599.89	−0.474041
$226$	0	0
$227$	2803.89	0.819828	0.409914	−	0.912124i	$-0.365559\pi$
$227$	2803.89	0.819828	0.409914	+	0.912124i	$0.365559\pi$
$228$	0	0
$229$	−1230.30	−0.355023	−0.177511	−	0.984119i	$-0.556805\pi$
$229$	−1230.30	−0.355023	−0.177511	+	0.984119i	$0.556805\pi$
$230$	0	0
$231$	−1125.52	−0.320578
$232$	0	0
$233$	−3356.06	−0.943617	−0.471808	−	0.881701i	$-0.656399\pi$
$233$	−3356.06	−0.943617	−0.471808	+	0.881701i	$0.656399\pi$
$234$	0	0
$235$	7836.18	2.17522
$236$	0	0
$237$	2444.17	0.669899
$238$	0	0
$239$	5649.77	1.52909	0.764547	−	0.644568i	$-0.222963\pi$
$239$	5649.77	1.52909	0.764547	+	0.644568i	$0.222963\pi$
$240$	0	0
$241$	−4371.24	−1.16837	−0.584183	−	0.811622i	$-0.698585\pi$
$241$	−4371.24	−1.16837	−0.584183	+	0.811622i	$0.698585\pi$
$242$	0	0
$243$	3161.88	0.834710
$244$	0	0
$245$	−4140.32	−1.07965
$246$	0	0
$247$	6920.82	1.78284
$248$	0	0
$249$	−3238.71	−0.824276
$250$	0	0
$251$	6689.89	1.68232	0.841159	−	0.540788i	$-0.181874\pi$
$251$	6689.89	1.68232	0.841159	+	0.540788i	$0.181874\pi$
$252$	0	0
$253$	822.847	0.204474
$254$	0	0
$255$	−2426.97	−0.596010
$256$	0	0
$257$	1155.98	0.280577	0.140289	−	0.990111i	$-0.455197\pi$
$257$	1155.98	0.280577	0.140289	+	0.990111i	$0.455197\pi$
$258$	0	0
$259$	8283.03	1.98719
$260$	0	0
$261$	361.981	0.0858471
$262$	0	0
$263$	3590.66	0.841862	0.420931	−	0.907093i	$-0.361703\pi$
$263$	3590.66	0.841862	0.420931	+	0.907093i	$0.361703\pi$
$264$	0	0
$265$	−5089.03	−1.17969
$266$	0	0
$267$	−356.793	−0.0817805
$268$	0	0
$269$	3354.99	0.760435	0.380218	−	0.924897i	$-0.375849\pi$
$269$	3354.99	0.760435	0.380218	+	0.924897i	$0.375849\pi$
$270$	0	0
$271$	3256.16	0.729880	0.364940	−	0.931031i	$-0.381089\pi$
$271$	3256.16	0.729880	0.364940	+	0.931031i	$0.381089\pi$
$272$	0	0
$273$	−4684.43	−1.03852
$274$	0	0
$275$	1541.58	0.338040
$276$	0	0
$277$	2257.20	0.489611	0.244805	−	0.969572i	$-0.421276\pi$
$277$	2257.20	0.489611	0.244805	+	0.969572i	$0.421276\pi$
$278$	0	0
$279$	−329.074	−0.0706134
$280$	0	0
$281$	−2685.00	−0.570014	−0.285007	−	0.958525i	$-0.591996\pi$
$281$	−2685.00	−0.570014	−0.285007	+	0.958525i	$0.591996\pi$
$282$	0	0
$283$	−6747.88	−1.41739	−0.708693	−	0.705517i	$-0.750715\pi$
$283$	−6747.88	−1.41739	−0.708693	+	0.705517i	$0.750715\pi$
$284$	0	0
$285$	−8382.00	−1.74213
$286$	0	0
$287$	−11281.9	−2.32037
$288$	0	0
$289$	−3310.47	−0.673819
$290$	0	0
$291$	2823.14	0.568712
$292$	0	0
$293$	2576.44	0.513711	0.256855	−	0.966450i	$-0.417314\pi$
$293$	2576.44	0.513711	0.256855	+	0.966450i	$0.417314\pi$
$294$	0	0
$295$	4533.98	0.894842
$296$	0	0
$297$	−1809.33	−0.353495
$298$	0	0
$299$	3424.72	0.662396
$300$	0	0
$301$	1581.08	0.302764
$302$	0	0
$303$	−1747.98	−0.331415
$304$	0	0
$305$	−5200.35	−0.976299
$306$	0	0
$307$	7376.02	1.37124	0.685622	−	0.727958i	$-0.259530\pi$
$307$	7376.02	1.37124	0.685622	+	0.727958i	$0.259530\pi$
$308$	0	0
$309$	−1991.13	−0.366574
$310$	0	0
$311$	−6462.63	−1.17834	−0.589168	−	0.808011i	$-0.700544\pi$
$311$	−6462.63	−1.17834	−0.589168	+	0.808011i	$0.700544\pi$
$312$	0	0
$313$	2690.76	0.485912	0.242956	−	0.970037i	$-0.421883\pi$
$313$	2690.76	0.485912	0.242956	+	0.970037i	$0.421883\pi$
$314$	0	0
$315$	−4877.89	−0.872502
$316$	0	0
$317$	−1470.14	−0.260477	−0.130239	−	0.991483i	$-0.541574\pi$
$317$	−1470.14	−0.260477	−0.130239	+	0.991483i	$0.541574\pi$
$318$	0	0
$319$	−348.790	−0.0612178
$320$	0	0
$321$	5226.16	0.908710
$322$	0	0
$323$	5534.63	0.953421
$324$	0	0
$325$	6416.12	1.09508
$326$	0	0
$327$	−6110.44	−1.03336
$328$	0	0
$329$	12095.6	2.02691
$330$	0	0
$331$	−6442.87	−1.06989	−0.534943	−	0.844888i	$-0.679667\pi$
$331$	−6442.87	−1.06989	−0.534943	+	0.844888i	$0.679667\pi$
$332$	0	0
$333$	4209.63	0.692751
$334$	0	0
$335$	5187.71	0.846074
$336$	0	0
$337$	−4640.75	−0.750141	−0.375071	−	0.926996i	$-0.622382\pi$
$337$	−4640.75	−0.750141	−0.375071	+	0.926996i	$0.622382\pi$
$338$	0	0
$339$	−7489.39	−1.19991
$340$	0	0
$341$	317.082	0.0503547
$342$	0	0
$343$	2033.36	0.320090
$344$	0	0
$345$	−4147.77	−0.647271
$346$	0	0
$347$	−9404.71	−1.45496	−0.727480	−	0.686129i	$-0.759309\pi$
$347$	−9404.71	−1.45496	−0.727480	+	0.686129i	$0.759309\pi$
$348$	0	0
$349$	6258.96	0.959984	0.479992	−	0.877273i	$-0.340640\pi$
$349$	6258.96	0.959984	0.479992	+	0.877273i	$0.340640\pi$
$350$	0	0
$351$	−7530.49	−1.14515
$352$	0	0
$353$	−5051.74	−0.761691	−0.380846	−	0.924639i	$-0.624367\pi$
$353$	−5051.74	−0.761691	−0.380846	+	0.924639i	$0.624367\pi$
$354$	0	0
$355$	−1498.43	−0.224024
$356$	0	0
$357$	−3746.18	−0.555374
$358$	0	0
$359$	−4138.51	−0.608418	−0.304209	−	0.952605i	$-0.598392\pi$
$359$	−4138.51	−0.608418	−0.304209	+	0.952605i	$0.598392\pi$
$360$	0	0
$361$	12255.9	1.78684
$362$	0	0
$363$	−4520.25	−0.653586
$364$	0	0
$365$	−2135.87	−0.306291
$366$	0	0
$367$	9804.50	1.39452	0.697262	−	0.716816i	$-0.254401\pi$
$367$	9804.50	1.39452	0.697262	+	0.716816i	$0.254401\pi$
$368$	0	0
$369$	−5733.70	−0.808901
$370$	0	0
$371$	−7855.24	−1.09925
$372$	0	0
$373$	−8696.49	−1.20720	−0.603602	−	0.797286i	$-0.706268\pi$
$373$	−8696.49	−1.20720	−0.603602	+	0.797286i	$0.706268\pi$
$374$	0	0
$375$	−192.455	−0.0265022
$376$	0	0
$377$	−1451.67	−0.198316
$378$	0	0
$379$	1920.25	0.260256	0.130128	−	0.991497i	$-0.458461\pi$
$379$	1920.25	0.260256	0.130128	+	0.991497i	$0.458461\pi$
$380$	0	0
$381$	7531.81	1.01277
$382$	0	0
$383$	7469.15	0.996490	0.498245	−	0.867036i	$-0.333978\pi$
$383$	7469.15	0.996490	0.498245	+	0.867036i	$0.333978\pi$
$384$	0	0
$385$	4700.13	0.622184
$386$	0	0
$387$	803.540	0.105546
$388$	0	0
$389$	1973.60	0.257237	0.128619	−	0.991694i	$-0.458946\pi$
$389$	1973.60	0.257237	0.128619	+	0.991694i	$0.458946\pi$
$390$	0	0
$391$	2738.77	0.354234
$392$	0	0
$393$	−8159.32	−1.04729
$394$	0	0
$395$	−10206.8	−1.30015
$396$	0	0
$397$	−14777.0	−1.86810	−0.934050	−	0.357143i	$-0.883751\pi$
$397$	−14777.0	−1.86810	−0.934050	+	0.357143i	$0.883751\pi$
$398$	0	0
$399$	−12938.1	−1.62335
$400$	0	0
$401$	−1788.68	−0.222749	−0.111375	−	0.993778i	$-0.535525\pi$
$401$	−1788.68	−0.222749	−0.111375	+	0.993778i	$0.535525\pi$
$402$	0	0
$403$	1319.70	0.163124
$404$	0	0
$405$	3757.96	0.461073
$406$	0	0
$407$	−4056.22	−0.494003
$408$	0	0
$409$	−11240.0	−1.35888	−0.679442	−	0.733729i	$-0.737778\pi$
$409$	−11240.0	−1.35888	−0.679442	+	0.733729i	$0.737778\pi$
$410$	0	0
$411$	7345.91	0.881623
$412$	0	0
$413$	6998.48	0.833832
$414$	0	0
$415$	13524.8	1.59977
$416$	0	0
$417$	−6289.90	−0.738652
$418$	0	0
$419$	7402.44	0.863086	0.431543	−	0.902092i	$-0.357969\pi$
$419$	7402.44	0.863086	0.431543	+	0.902092i	$0.357969\pi$
$420$	0	0
$421$	8604.54	0.996104	0.498052	−	0.867147i	$-0.334049\pi$
$421$	8604.54	0.996104	0.498052	+	0.867147i	$0.334049\pi$
$422$	0	0
$423$	6147.28	0.706598
$424$	0	0
$425$	5131.02	0.585626
$426$	0	0
$427$	−8027.06	−0.909735
$428$	0	0
$429$	2293.98	0.258168
$430$	0	0
$431$	−10544.3	−1.17843	−0.589213	−	0.807978i	$-0.700562\pi$
$431$	−10544.3	−1.17843	−0.589213	+	0.807978i	$0.700562\pi$
$432$	0	0
$433$	−1910.13	−0.211998	−0.105999	−	0.994366i	$-0.533804\pi$
$433$	−1910.13	−0.211998	−0.105999	+	0.994366i	$0.533804\pi$
$434$	0	0
$435$	1758.16	0.193787
$436$	0	0
$437$	9458.87	1.03542
$438$	0	0
$439$	−7268.58	−0.790229	−0.395115	−	0.918632i	$-0.629295\pi$
$439$	−7268.58	−0.790229	−0.395115	+	0.918632i	$0.629295\pi$
$440$	0	0
$441$	−3247.97	−0.350715
$442$	0	0
$443$	−18328.3	−1.96569	−0.982847	−	0.184423i	$-0.940959\pi$
$443$	−18328.3	−1.96569	−0.982847	+	0.184423i	$0.940959\pi$
$444$	0	0
$445$	1489.96	0.158721
$446$	0	0
$447$	−7435.44	−0.786766
$448$	0	0
$449$	−1802.68	−0.189474	−0.0947371	−	0.995502i	$-0.530201\pi$
$449$	−1802.68	−0.189474	−0.0947371	+	0.995502i	$0.530201\pi$
$450$	0	0
$451$	5524.75	0.576830
$452$	0	0
$453$	−9216.45	−0.955908
$454$	0	0
$455$	19562.1	2.01557
$456$	0	0
$457$	−1118.56	−0.114494	−0.0572472	−	0.998360i	$-0.518232\pi$
$457$	−1118.56	−0.114494	−0.0572472	+	0.998360i	$0.518232\pi$
$458$	0	0
$459$	−6022.19	−0.612400
$460$	0	0
$461$	−1826.61	−0.184542	−0.0922708	−	0.995734i	$-0.529413\pi$
$461$	−1826.61	−0.184542	−0.0922708	+	0.995734i	$0.529413\pi$
$462$	0	0
$463$	18162.3	1.82305	0.911525	−	0.411245i	$-0.134906\pi$
$463$	18162.3	1.82305	0.911525	+	0.411245i	$0.134906\pi$
$464$	0	0
$465$	−1598.33	−0.159400
$466$	0	0
$467$	3096.21	0.306799	0.153400	−	0.988164i	$-0.450978\pi$
$467$	3096.21	0.306799	0.153400	+	0.988164i	$0.450978\pi$
$468$	0	0
$469$	8007.55	0.788389
$470$	0	0
$471$	−12598.2	−1.23247
$472$	0	0
$473$	−774.257	−0.0752651
$474$	0	0
$475$	17721.0	1.71178
$476$	0	0
$477$	−3992.21	−0.383209
$478$	0	0
$479$	4.10178	0.000391263 0	0.000195632	−	1.00000i	$-0.499938\pi$
$479$	4.10178	0.000391263 0	0.000195632		1.00000i	$0.499938\pi$
$480$	0	0
$481$	−16882.1	−1.60033
$482$	0	0
$483$	−6402.34	−0.603140
$484$	0	0
$485$	−11789.4	−1.10377
$486$	0	0
$487$	−608.258	−0.0565971	−0.0282986	−	0.999600i	$-0.509009\pi$
$487$	−608.258	−0.0565971	−0.0282986	+	0.999600i	$0.509009\pi$
$488$	0	0
$489$	7641.20	0.706640
$490$	0	0
$491$	−11528.6	−1.05963	−0.529813	−	0.848114i	$-0.677738\pi$
$491$	−11528.6	−1.05963	−0.529813	+	0.848114i	$0.677738\pi$
$492$	0	0
$493$	−1160.92	−0.106055
$494$	0	0
$495$	2388.71	0.216898
$496$	0	0
$497$	−2312.92	−0.208750
$498$	0	0
$499$	15038.8	1.34916	0.674580	−	0.738202i	$-0.264325\pi$
$499$	15038.8	1.34916	0.674580	+	0.738202i	$0.264325\pi$
$500$	0	0
$501$	8946.45	0.797801
$502$	0	0
$503$	11376.3	1.00844	0.504220	−	0.863575i	$-0.331780\pi$
$503$	11376.3	1.00844	0.504220	+	0.863575i	$0.331780\pi$
$504$	0	0
$505$	7299.51	0.643216
$506$	0	0
$507$	1176.52	0.103059
$508$	0	0
$509$	10134.0	0.882476	0.441238	−	0.897390i	$-0.354540\pi$
$509$	10134.0	0.882476	0.441238	+	0.897390i	$0.354540\pi$
$510$	0	0
$511$	−3296.84	−0.285408
$512$	0	0
$513$	−20798.8	−1.79004
$514$	0	0
$515$	8314.91	0.711454
$516$	0	0
$517$	−5923.26	−0.503877
$518$	0	0
$519$	−215.170	−0.0181983
$520$	0	0
$521$	−19299.6	−1.62290	−0.811449	−	0.584423i	$-0.801321\pi$
$521$	−19299.6	−1.62290	−0.811449	+	0.584423i	$0.801321\pi$
$522$	0	0
$523$	3730.87	0.311931	0.155965	−	0.987763i	$-0.450151\pi$
$523$	3730.87	0.311931	0.155965	+	0.987763i	$0.450151\pi$
$524$	0	0
$525$	−11994.6	−0.997122
$526$	0	0
$527$	1055.38	0.0872352
$528$	0	0
$529$	−7486.35	−0.615299
$530$	0	0
$531$	3556.79	0.290681
$532$	0	0
$533$	22994.2	1.86865
$534$	0	0
$535$	−21824.3	−1.76364
$536$	0	0
$537$	−9977.61	−0.801798
$538$	0	0
$539$	3129.61	0.250096
$540$	0	0
$541$	−21669.3	−1.72206	−0.861030	−	0.508554i	$-0.830180\pi$
$541$	−21669.3	−1.72206	−0.861030	+	0.508554i	$0.830180\pi$
$542$	0	0
$543$	−11679.0	−0.923008
$544$	0	0
$545$	25517.1	2.00556
$546$	0	0
$547$	−10124.8	−0.791417	−0.395709	−	0.918376i	$-0.629501\pi$
$547$	−10124.8	−0.791417	−0.395709	+	0.918376i	$0.629501\pi$
$548$	0	0
$549$	−4079.54	−0.317141
$550$	0	0
$551$	−4009.44	−0.309996
$552$	0	0
$553$	−15754.9	−1.21151
$554$	0	0
$555$	20446.4	1.56379
$556$	0	0
$557$	−15890.2	−1.20878	−0.604390	−	0.796689i	$-0.706583\pi$
$557$	−15890.2	−1.20878	−0.604390	+	0.796689i	$0.706583\pi$
$558$	0	0
$559$	−3222.48	−0.243822
$560$	0	0
$561$	1834.51	0.138063
$562$	0	0
$563$	−16494.3	−1.23473	−0.617366	−	0.786676i	$-0.711800\pi$
$563$	−16494.3	−1.23473	−0.617366	+	0.786676i	$0.711800\pi$
$564$	0	0
$565$	31275.5	2.32880
$566$	0	0
$567$	5800.65	0.429637
$568$	0	0
$569$	18593.7	1.36993	0.684965	−	0.728576i	$-0.259817\pi$
$569$	18593.7	1.36993	0.684965	+	0.728576i	$0.259817\pi$
$570$	0	0
$571$	−3943.82	−0.289044	−0.144522	−	0.989502i	$-0.546164\pi$
$571$	−3943.82	−0.289044	−0.144522	+	0.989502i	$0.546164\pi$
$572$	0	0
$573$	−11214.5	−0.817611
$574$	0	0
$575$	8769.09	0.635994
$576$	0	0
$577$	−19327.2	−1.39445	−0.697227	−	0.716850i	$-0.745583\pi$
$577$	−19327.2	−1.39445	−0.697227	+	0.716850i	$0.745583\pi$
$578$	0	0
$579$	544.258	0.0390649
$580$	0	0
$581$	20876.3	1.49070
$582$	0	0
$583$	3846.72	0.273268
$584$	0	0
$585$	9941.90	0.702645
$586$	0	0
$587$	2406.26	0.169194	0.0845970	−	0.996415i	$-0.473040\pi$
$587$	2406.26	0.169194	0.0845970	+	0.996415i	$0.473040\pi$
$588$	0	0
$589$	3644.95	0.254987
$590$	0	0
$591$	−2806.51	−0.195337
$592$	0	0
$593$	23175.1	1.60487	0.802435	−	0.596740i	$-0.203538\pi$
$593$	23175.1	1.60487	0.802435	+	0.596740i	$0.203538\pi$
$594$	0	0
$595$	15644.0	1.07788
$596$	0	0
$597$	12579.2	0.862368
$598$	0	0
$599$	−13870.8	−0.946150	−0.473075	−	0.881022i	$-0.656856\pi$
$599$	−13870.8	−0.946150	−0.473075	+	0.881022i	$0.656856\pi$
$600$	0	0
$601$	−11975.4	−0.812787	−0.406393	−	0.913698i	$-0.633214\pi$
$601$	−11975.4	−0.812787	−0.406393	+	0.913698i	$0.633214\pi$
$602$	0	0
$603$	4069.62	0.274839
$604$	0	0
$605$	18876.5	1.26849
$606$	0	0
$607$	11549.3	0.772279	0.386139	−	0.922440i	$-0.373808\pi$
$607$	11549.3	0.772279	0.386139	+	0.922440i	$0.373808\pi$
$608$	0	0
$609$	2713.84	0.180575
$610$	0	0
$611$	−24652.8	−1.63232
$612$	0	0
$613$	−22900.2	−1.50886	−0.754431	−	0.656380i	$-0.772087\pi$
$613$	−22900.2	−1.50886	−0.754431	+	0.656380i	$0.772087\pi$
$614$	0	0
$615$	−27848.9	−1.82598
$616$	0	0
$617$	−24950.5	−1.62799	−0.813996	−	0.580871i	$-0.802712\pi$
$617$	−24950.5	−1.62799	−0.813996	+	0.580871i	$0.802712\pi$
$618$	0	0
$619$	1861.09	0.120846	0.0604228	−	0.998173i	$-0.480755\pi$
$619$	1861.09	0.120846	0.0604228	+	0.998173i	$0.480755\pi$
$620$	0	0
$621$	−10292.1	−0.665071
$622$	0	0
$623$	2299.85	0.147900
$624$	0	0
$625$	−15218.1	−0.973960
$626$	0	0
$627$	6335.83	0.403555
$628$	0	0
$629$	−13500.7	−0.855819
$630$	0	0
$631$	−15428.2	−0.973357	−0.486679	−	0.873581i	$-0.661792\pi$
$631$	−15428.2	−0.973357	−0.486679	+	0.873581i	$0.661792\pi$
$632$	0	0
$633$	−12703.2	−0.797642
$634$	0	0
$635$	−31452.7	−1.96561
$636$	0	0
$637$	13025.5	0.810188
$638$	0	0
$639$	−1175.48	−0.0727719
$640$	0	0
$641$	22934.5	1.41319	0.706597	−	0.707616i	$-0.250229\pi$
$641$	22934.5	1.41319	0.706597	+	0.707616i	$0.250229\pi$
$642$	0	0
$643$	−25079.4	−1.53816	−0.769079	−	0.639154i	$-0.779285\pi$
$643$	−25079.4	−1.53816	−0.769079	+	0.639154i	$0.779285\pi$
$644$	0	0
$645$	3902.84	0.238254
$646$	0	0
$647$	24223.2	1.47189	0.735944	−	0.677042i	$-0.236739\pi$
$647$	24223.2	1.47189	0.735944	+	0.677042i	$0.236739\pi$
$648$	0	0
$649$	−3427.17	−0.207285
$650$	0	0
$651$	−2467.12	−0.148532
$652$	0	0
$653$	−9345.95	−0.560084	−0.280042	−	0.959988i	$-0.590348\pi$
$653$	−9345.95	−0.560084	−0.280042	+	0.959988i	$0.590348\pi$
$654$	0	0
$655$	34073.1	2.03259
$656$	0	0
$657$	−1675.53	−0.0994957
$658$	0	0
$659$	27391.6	1.61916	0.809579	−	0.587011i	$-0.199696\pi$
$659$	27391.6	1.61916	0.809579	+	0.587011i	$0.199696\pi$
$660$	0	0
$661$	12893.1	0.758672	0.379336	−	0.925259i	$-0.376153\pi$
$661$	12893.1	0.758672	0.379336	+	0.925259i	$0.376153\pi$
$662$	0	0
$663$	7635.29	0.447255
$664$	0	0
$665$	54029.4	3.15063
$666$	0	0
$667$	−1984.04	−0.115176
$668$	0	0
$669$	−5979.19	−0.345544
$670$	0	0
$671$	3930.87	0.226154
$672$	0	0
$673$	−5210.29	−0.298428	−0.149214	−	0.988805i	$-0.547674\pi$
$673$	−5210.29	−0.298428	−0.149214	+	0.988805i	$0.547674\pi$
$674$	0	0
$675$	−19282.1	−1.09951
$676$	0	0
$677$	−18546.3	−1.05287	−0.526435	−	0.850215i	$-0.676472\pi$
$677$	−18546.3	−1.05287	−0.526435	+	0.850215i	$0.676472\pi$
$678$	0	0
$679$	−18197.6	−1.02851
$680$	0	0
$681$	10683.5	0.601163
$682$	0	0
$683$	17390.1	0.974252	0.487126	−	0.873332i	$-0.338045\pi$
$683$	17390.1	0.974252	0.487126	+	0.873332i	$0.338045\pi$
$684$	0	0
$685$	−30676.4	−1.71107
$686$	0	0
$687$	−4687.71	−0.260331
$688$	0	0
$689$	16010.2	0.885253
$690$	0	0
$691$	25921.0	1.42703	0.713516	−	0.700639i	$-0.247101\pi$
$691$	25921.0	1.42703	0.713516	+	0.700639i	$0.247101\pi$
$692$	0	0
$693$	3687.13	0.202110
$694$	0	0
$695$	26266.5	1.43359
$696$	0	0
$697$	18388.6	0.999309
$698$	0	0
$699$	−12787.4	−0.691935
$700$	0	0
$701$	−5372.45	−0.289465	−0.144732	−	0.989471i	$-0.546232\pi$
$701$	−5372.45	−0.289465	−0.144732	+	0.989471i	$0.546232\pi$
$702$	0	0
$703$	−46627.4	−2.50155
$704$	0	0
$705$	29857.7	1.59504
$706$	0	0
$707$	11267.3	0.599362
$708$	0	0
$709$	−8160.24	−0.432249	−0.216124	−	0.976366i	$-0.569342\pi$
$709$	−8160.24	−0.432249	−0.216124	+	0.976366i	$0.569342\pi$
$710$	0	0
$711$	−8006.98	−0.422342
$712$	0	0
$713$	1803.68	0.0947380
$714$	0	0
$715$	−9579.60	−0.501058
$716$	0	0
$717$	21526.9	1.12125
$718$	0	0
$719$	18890.0	0.979803	0.489902	−	0.871778i	$-0.337033\pi$
$719$	18890.0	0.979803	0.489902	+	0.871778i	$0.337033\pi$
$720$	0	0
$721$	12834.6	0.662948
$722$	0	0
$723$	−16655.4	−0.856739
$724$	0	0
$725$	−3717.06	−0.190411
$726$	0	0
$727$	11715.9	0.597689	0.298844	−	0.954302i	$-0.403399\pi$
$727$	11715.9	0.597689	0.298844	+	0.954302i	$0.403399\pi$
$728$	0	0
$729$	18424.3	0.936053
$730$	0	0
$731$	−2577.04	−0.130390
$732$	0	0
$733$	13436.3	0.677053	0.338526	−	0.940957i	$-0.390072\pi$
$733$	13436.3	0.677053	0.338526	+	0.940957i	$0.390072\pi$
$734$	0	0
$735$	−15775.6	−0.791688
$736$	0	0
$737$	−3921.32	−0.195988
$738$	0	0
$739$	−35978.3	−1.79091	−0.895455	−	0.445151i	$-0.853150\pi$
$739$	−35978.3	−1.79091	−0.895455	+	0.445151i	$0.853150\pi$
$740$	0	0
$741$	26369.9	1.30732
$742$	0	0
$743$	−24156.6	−1.19276	−0.596380	−	0.802702i	$-0.703395\pi$
$743$	−24156.6	−1.19276	−0.596380	+	0.802702i	$0.703395\pi$
$744$	0	0
$745$	31050.2	1.52697
$746$	0	0
$747$	10609.8	0.519670
$748$	0	0
$749$	−33687.2	−1.64340
$750$	0	0
$751$	−19167.6	−0.931338	−0.465669	−	0.884959i	$-0.654186\pi$
$751$	−19167.6	−0.931338	−0.465669	+	0.884959i	$0.654186\pi$
$752$	0	0
$753$	25490.0	1.23361
$754$	0	0
$755$	38487.7	1.85525
$756$	0	0
$757$	40535.4	1.94622	0.973108	−	0.230350i	$-0.0739870\pi$
$757$	40535.4	1.94622	0.973108	+	0.230350i	$0.0739870\pi$
$758$	0	0
$759$	3135.24	0.149937
$760$	0	0
$761$	24492.1	1.16667	0.583336	−	0.812231i	$-0.301747\pi$
$761$	24492.1	1.16667	0.583336	+	0.812231i	$0.301747\pi$
$762$	0	0
$763$	39387.2	1.86882
$764$	0	0
$765$	7950.61	0.375758
$766$	0	0
$767$	−14264.0	−0.671503
$768$	0	0
$769$	1732.55	0.0812448	0.0406224	−	0.999175i	$-0.487066\pi$
$769$	1732.55	0.0812448	0.0406224	+	0.999175i	$0.487066\pi$
$770$	0	0
$771$	4404.57	0.205742
$772$	0	0
$773$	−28668.3	−1.33393	−0.666966	−	0.745089i	$-0.732407\pi$
$773$	−28668.3	−1.33393	−0.666966	+	0.745089i	$0.732407\pi$
$774$	0	0
$775$	3379.14	0.156622
$776$	0	0
$777$	31560.3	1.45717
$778$	0	0
$779$	63508.6	2.92097
$780$	0	0
$781$	1132.64	0.0518939
$782$	0	0
$783$	4362.65	0.199117
$784$	0	0
$785$	52609.8	2.39200
$786$	0	0
$787$	2751.31	0.124617	0.0623085	−	0.998057i	$-0.480154\pi$
$787$	2751.31	0.124617	0.0623085	+	0.998057i	$0.480154\pi$
$788$	0	0
$789$	13681.3	0.617320
$790$	0	0
$791$	48275.7	2.17002
$792$	0	0
$793$	16360.4	0.732629
$794$	0	0
$795$	−19390.4	−0.865039
$796$	0	0
$797$	−8410.25	−0.373784	−0.186892	−	0.982380i	$-0.559842\pi$
$797$	−8410.25	−0.373784	−0.186892	+	0.982380i	$0.559842\pi$
$798$	0	0
$799$	−19715.0	−0.872925
$800$	0	0
$801$	1168.84	0.0515590
$802$	0	0
$803$	1614.47	0.0709507
$804$	0	0
$805$	26736.1	1.17059
$806$	0	0
$807$	12783.3	0.557612
$808$	0	0
$809$	−9423.18	−0.409520	−0.204760	−	0.978812i	$-0.565641\pi$
$809$	−9423.18	−0.409520	−0.204760	+	0.978812i	$0.565641\pi$
$810$	0	0
$811$	−9836.18	−0.425888	−0.212944	−	0.977064i	$-0.568305\pi$
$811$	−9836.18	−0.425888	−0.212944	+	0.977064i	$0.568305\pi$
$812$	0	0
$813$	12406.7	0.535207
$814$	0	0
$815$	−31909.5	−1.37146
$816$	0	0
$817$	−8900.31	−0.381129
$818$	0	0
$819$	15346.0	0.654739
$820$	0	0
$821$	30153.1	1.28179	0.640896	−	0.767628i	$-0.278563\pi$
$821$	30153.1	1.28179	0.640896	+	0.767628i	$0.278563\pi$
$822$	0	0
$823$	−23948.3	−1.01432	−0.507161	−	0.861852i	$-0.669305\pi$
$823$	−23948.3	−1.01432	−0.507161	+	0.861852i	$0.669305\pi$
$824$	0	0
$825$	5873.80	0.247878
$826$	0	0
$827$	−7854.32	−0.330256	−0.165128	−	0.986272i	$-0.552804\pi$
$827$	−7854.32	−0.330256	−0.165128	+	0.986272i	$0.552804\pi$
$828$	0	0
$829$	22496.0	0.942483	0.471242	−	0.882004i	$-0.343806\pi$
$829$	22496.0	0.942483	0.471242	+	0.882004i	$0.343806\pi$
$830$	0	0
$831$	8600.47	0.359022
$832$	0	0
$833$	10416.6	0.433270
$834$	0	0
$835$	−37360.2	−1.54839
$836$	0	0
$837$	−3966.04	−0.163783
$838$	0	0
$839$	35183.4	1.44776	0.723878	−	0.689928i	$-0.242358\pi$
$839$	35183.4	1.44776	0.723878	+	0.689928i	$0.242358\pi$
$840$	0	0
$841$	841.000	0.0344828
$842$	0	0
$843$	−10230.5	−0.417980
$844$	0	0
$845$	−4913.11	−0.200019
$846$	0	0
$847$	29137.0	1.18201
$848$	0	0
$849$	−25711.0	−1.03934
$850$	0	0
$851$	−23073.2	−0.929425
$852$	0	0
$853$	44588.5	1.78978	0.894889	−	0.446289i	$-0.147255\pi$
$853$	44588.5	1.78978	0.894889	+	0.446289i	$0.147255\pi$
$854$	0	0
$855$	27459.0	1.09834
$856$	0	0
$857$	29852.1	1.18988	0.594941	−	0.803769i	$-0.297175\pi$
$857$	29852.1	1.18988	0.594941	+	0.803769i	$0.297175\pi$
$858$	0	0
$859$	−25597.6	−1.01674	−0.508370	−	0.861139i	$-0.669752\pi$
$859$	−25597.6	−1.01674	−0.508370	+	0.861139i	$0.669752\pi$
$860$	0	0
$861$	−42986.5	−1.70148
$862$	0	0
$863$	−7808.35	−0.307995	−0.153997	−	0.988071i	$-0.549215\pi$
$863$	−7808.35	−0.307995	−0.153997	+	0.988071i	$0.549215\pi$
$864$	0	0
$865$	898.544	0.0353195
$866$	0	0
$867$	−12613.7	−0.494098
$868$	0	0
$869$	7715.19	0.301173
$870$	0	0
$871$	−16320.6	−0.634906
$872$	0	0
$873$	−9248.45	−0.358548
$874$	0	0
$875$	1240.54	0.0479290
$876$	0	0
$877$	−26177.8	−1.00794	−0.503969	−	0.863722i	$-0.668127\pi$
$877$	−26177.8	−1.00794	−0.503969	+	0.863722i	$0.668127\pi$
$878$	0	0
$879$	9816.84	0.376694
$880$	0	0
$881$	−48456.0	−1.85304	−0.926519	−	0.376249i	$-0.877214\pi$
$881$	−48456.0	−1.85304	−0.926519	+	0.376249i	$0.877214\pi$
$882$	0	0
$883$	−14211.7	−0.541631	−0.270816	−	0.962631i	$-0.587293\pi$
$883$	−14211.7	−0.541631	−0.270816	+	0.962631i	$0.587293\pi$
$884$	0	0
$885$	17275.5	0.656170
$886$	0	0
$887$	−20489.4	−0.775612	−0.387806	−	0.921741i	$-0.626767\pi$
$887$	−20489.4	−0.775612	−0.387806	+	0.921741i	$0.626767\pi$
$888$	0	0
$889$	−48549.2	−1.83159
$890$	0	0
$891$	−2840.59	−0.106805
$892$	0	0
$893$	−68089.6	−2.55155
$894$	0	0
$895$	41666.3	1.55615
$896$	0	0
$897$	13049.0	0.485722
$898$	0	0
$899$	−764.545	−0.0283638
$900$	0	0
$901$	12803.5	0.473413
$902$	0	0
$903$	6024.28	0.222010
$904$	0	0
$905$	48771.2	1.79139
$906$	0	0
$907$	−33584.1	−1.22948	−0.614741	−	0.788729i	$-0.710740\pi$
$907$	−33584.1	−1.22948	−0.614741	+	0.788729i	$0.710740\pi$
$908$	0	0
$909$	5726.28	0.208942
$910$	0	0
$911$	−36834.4	−1.33960	−0.669802	−	0.742540i	$-0.733621\pi$
$911$	−36834.4	−1.33960	−0.669802	+	0.742540i	$0.733621\pi$
$912$	0	0
$913$	−10223.2	−0.370578
$914$	0	0
$915$	−19814.5	−0.715900
$916$	0	0
$917$	52594.0	1.89401
$918$	0	0
$919$	14301.5	0.513344	0.256672	−	0.966499i	$-0.417374\pi$
$919$	14301.5	0.513344	0.256672	+	0.966499i	$0.417374\pi$
$920$	0	0
$921$	28104.4	1.00551
$922$	0	0
$923$	4714.09	0.168111
$924$	0	0
$925$	−43227.2	−1.53654
$926$	0	0
$927$	6522.83	0.231109
$928$	0	0
$929$	21455.5	0.757733	0.378866	−	0.925451i	$-0.376314\pi$
$929$	21455.5	0.757733	0.378866	+	0.925451i	$0.376314\pi$
$930$	0	0
$931$	35975.7	1.26644
$932$	0	0
$933$	−24624.1	−0.864050
$934$	0	0
$935$	−7660.88	−0.267954
$936$	0	0
$937$	−13814.7	−0.481650	−0.240825	−	0.970569i	$-0.577418\pi$
$937$	−13814.7	−0.481650	−0.240825	+	0.970569i	$0.577418\pi$
$938$	0	0
$939$	10252.4	0.356310
$940$	0	0
$941$	−29884.0	−1.03527	−0.517636	−	0.855601i	$-0.673188\pi$
$941$	−29884.0	−1.03527	−0.517636	+	0.855601i	$0.673188\pi$
$942$	0	0
$943$	31426.8	1.08526
$944$	0	0
$945$	−58789.0	−2.02371
$946$	0	0
$947$	39549.4	1.35711	0.678554	−	0.734550i	$-0.262607\pi$
$947$	39549.4	1.35711	0.678554	+	0.734550i	$0.262607\pi$
$948$	0	0
$949$	6719.47	0.229845
$950$	0	0
$951$	−5601.57	−0.191003
$952$	0	0
$953$	−4627.44	−0.157290	−0.0786450	−	0.996903i	$-0.525059\pi$
$953$	−4627.44	−0.157290	−0.0786450	+	0.996903i	$0.525059\pi$
$954$	0	0
$955$	46831.4	1.58683
$956$	0	0
$957$	−1328.97	−0.0448898
$958$	0	0
$959$	−47350.9	−1.59441
$960$	0	0
$961$	−29096.0	−0.976669
$962$	0	0
$963$	−17120.6	−0.572902
$964$	0	0
$965$	−2272.81	−0.0758179
$966$	0	0
$967$	23922.8	0.795558	0.397779	−	0.917481i	$-0.369781\pi$
$967$	23922.8	0.795558	0.397779	+	0.917481i	$0.369781\pi$
$968$	0	0
$969$	21088.2	0.699125
$970$	0	0
$971$	41189.3	1.36131	0.680653	−	0.732606i	$-0.261696\pi$
$971$	41189.3	1.36131	0.680653	+	0.732606i	$0.261696\pi$
$972$	0	0
$973$	40544.0	1.33585
$974$	0	0
$975$	24446.9	0.803003
$976$	0	0
$977$	23579.2	0.772125	0.386063	−	0.922473i	$-0.373835\pi$
$977$	23579.2	0.772125	0.386063	+	0.922473i	$0.373835\pi$
$978$	0	0
$979$	−1126.24	−0.0367669
$980$	0	0
$981$	20017.5	0.651487
$982$	0	0
$983$	10073.3	0.326846	0.163423	−	0.986556i	$-0.447747\pi$
$983$	10073.3	0.326846	0.163423	+	0.986556i	$0.447747\pi$
$984$	0	0
$985$	11719.9	0.379115
$986$	0	0
$987$	46087.2	1.48629
$988$	0	0
$989$	−4404.25	−0.141605
$990$	0	0
$991$	−10597.4	−0.339695	−0.169847	−	0.985470i	$-0.554327\pi$
$991$	−10597.4	−0.339695	−0.169847	+	0.985470i	$0.554327\pi$
$992$	0	0
$993$	−24548.8	−0.784525
$994$	0	0
$995$	−52530.6	−1.67370
$996$	0	0
$997$	27251.7	0.865666	0.432833	−	0.901474i	$-0.357514\pi$
$997$	27251.7	0.865666	0.432833	+	0.901474i	$0.357514\pi$
$998$	0	0
$999$	50735.0	1.60679

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1856.4.a.bh.1.7		10
4.3	odd	2	inner	1856.4.a.bh.1.4		10
8.3	odd	2		928.4.a.g.1.7	yes	10
8.5	even	2		928.4.a.g.1.4	✓	10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
928.4.a.g.1.4	✓	10	8.5	even	2
928.4.a.g.1.7	yes	10	8.3	odd	2
1856.4.a.bh.1.4		10	4.3	odd	2	inner
1856.4.a.bh.1.7		10	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1856 = 2^{6} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1856.a (trivial)

\(f(q)\)	\(=\)	\(q+3.81023 q^{3} -15.9115 q^{5} -24.5603 q^{7} -12.4821 q^{9} +O(q^{10})\)	\(q+3.81023 q^{3} -15.9115 q^{5} -24.5603 q^{7} -12.4821 q^{9} +12.0272 q^{11} +50.0577 q^{13} -60.6264 q^{15} +40.0316 q^{17} +138.257 q^{19} -93.5806 q^{21} +68.4153 q^{23} +128.174 q^{25} -150.436 q^{27} -29.0000 q^{29} +26.3636 q^{31} +45.8266 q^{33} +390.791 q^{35} -337.253 q^{37} +190.732 q^{39} +459.353 q^{41} -64.3753 q^{43} +198.609 q^{45} -492.487 q^{47} +260.210 q^{49} +152.530 q^{51} +319.834 q^{53} -191.371 q^{55} +526.790 q^{57} -284.951 q^{59} +326.831 q^{61} +306.565 q^{63} -796.492 q^{65} -326.036 q^{67} +260.678 q^{69} +94.1730 q^{71} +134.234 q^{73} +488.375 q^{75} -295.393 q^{77} +641.476 q^{79} -236.179 q^{81} -850.002 q^{83} -636.960 q^{85} -110.497 q^{87} -93.6408 q^{89} -1229.43 q^{91} +100.452 q^{93} -2199.87 q^{95} +740.936 q^{97} -150.125 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 40 q^{5} + 58 q^{9}+O(q^{10}) \) 10 * q - 40 * q^5 + 58 * q^9	\( 10 q - 40 q^{5} + 58 q^{9} - 136 q^{13} + 144 q^{17} - 92 q^{21} + 334 q^{25} - 290 q^{29} + 536 q^{33} - 212 q^{37} + 400 q^{41} - 748 q^{45} + 626 q^{49} - 1304 q^{53} + 408 q^{57} - 1600 q^{61} + 1808 q^{65} - 1916 q^{69} + 1332 q^{73} - 1724 q^{77} + 730 q^{81} - 3484 q^{85} + 2256 q^{89} - 3500 q^{93} + 560 q^{97}+O(q^{100}) \) 10 * q - 40 * q^5 + 58 * q^9 - 136 * q^13 + 144 * q^17 - 92 * q^21 + 334 * q^25 - 290 * q^29 + 536 * q^33 - 212 * q^37 + 400 * q^41 - 748 * q^45 + 626 * q^49 - 1304 * q^53 + 408 * q^57 - 1600 * q^61 + 1808 * q^65 - 1916 * q^69 + 1332 * q^73 - 1724 * q^77 + 730 * q^81 - 3484 * q^85 + 2256 * q^89 - 3500 * q^93 + 560 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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