LMFDB - Embedded newform 1856.4.a.bk.1.9

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:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 245x^{10} + 21294x^{8} - 755514x^{6} + 8955005x^{4} - 27099393x^{2} + 23710340 $ x^12 - 245x^10 + 21294x^8 - 755514x^6 + 8955005x^4 - 27099393*x^2 + 23710340
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{19} $
Twist minimal:	no (minimal twist has level 928)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.9
Root			$3.82226$ 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)

$f(q)$	$=$	$q+3.82226 q^{3} -3.68395 q^{5} -23.1652 q^{7} -12.3903 q^{9} +O(q^{10})$	$q+3.82226 q^{3} -3.68395 q^{5} -23.1652 q^{7} -12.3903 q^{9} +16.6789 q^{11} -87.9352 q^{13} -14.0810 q^{15} -58.1595 q^{17} +102.608 q^{19} -88.5436 q^{21} -111.738 q^{23} -111.429 q^{25} -150.560 q^{27} +29.0000 q^{29} +146.296 q^{31} +63.7510 q^{33} +85.3395 q^{35} +77.3874 q^{37} -336.111 q^{39} +245.696 q^{41} +369.510 q^{43} +45.6454 q^{45} -3.90706 q^{47} +193.629 q^{49} -222.301 q^{51} -355.978 q^{53} -61.4441 q^{55} +392.196 q^{57} -223.920 q^{59} +308.765 q^{61} +287.025 q^{63} +323.948 q^{65} -548.837 q^{67} -427.091 q^{69} -121.875 q^{71} +228.283 q^{73} -425.909 q^{75} -386.371 q^{77} +665.320 q^{79} -240.940 q^{81} +1417.96 q^{83} +214.256 q^{85} +110.845 q^{87} +955.874 q^{89} +2037.04 q^{91} +559.179 q^{93} -378.004 q^{95} -598.996 q^{97} -206.657 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 10 q^{5} + 166 q^{9}+O(q^{10}) $ 12 * q - 10 * q^5 + 166 * q^9	$ 12 q - 10 q^{5} + 166 q^{9} - 30 q^{13} + 336 q^{17} - 56 q^{21} + 294 q^{25} + 348 q^{29} + 214 q^{33} - 196 q^{37} + 940 q^{41} - 1672 q^{45} + 972 q^{49} + 406 q^{53} + 1224 q^{57} - 400 q^{61} + 2998 q^{65} - 1004 q^{69} + 2252 q^{73} - 3032 q^{77} + 3932 q^{81} - 5564 q^{85} + 5212 q^{89} - 3722 q^{93} + 6164 q^{97}+O(q^{100}) $ 12 * q - 10 * q^5 + 166 * q^9 - 30 * q^13 + 336 * q^17 - 56 * q^21 + 294 * q^25 + 348 * q^29 + 214 * q^33 - 196 * q^37 + 940 * q^41 - 1672 * q^45 + 972 * q^49 + 406 * q^53 + 1224 * q^57 - 400 * q^61 + 2998 * q^65 - 1004 * q^69 + 2252 * q^73 - 3032 * q^77 + 3932 * q^81 - 5564 * q^85 + 5212 * q^89 - 3722 * q^93 + 6164 * 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
$3$	3.82226	0.735594	0.367797	−	0.929906i	$-0.380112\pi$
$3$	3.82226	0.735594	0.367797	+	0.929906i	$0.380112\pi$
$4$	0	0
$5$	−3.68395	−0.329502	−0.164751	−	0.986335i	$-0.552682\pi$
$5$	−3.68395	−0.329502	−0.164751	+	0.986335i	$0.552682\pi$
$6$	0	0
$7$	−23.1652	−1.25081	−0.625403	−	0.780302i	$-0.715065\pi$
$7$	−23.1652	−1.25081	−0.625403	+	0.780302i	$0.715065\pi$
$8$	0	0
$9$	−12.3903	−0.458902
$10$	0	0
$11$	16.6789	0.457170	0.228585	−	0.973524i	$-0.426590\pi$
$11$	16.6789	0.457170	0.228585	+	0.973524i	$0.426590\pi$
$12$	0	0
$13$	−87.9352	−1.87606	−0.938032	−	0.346549i	$-0.887354\pi$
$13$	−87.9352	−1.87606	−0.938032	+	0.346549i	$0.887354\pi$
$14$	0	0
$15$	−14.0810	−0.242380
$16$	0	0
$17$	−58.1595	−0.829750	−0.414875	−	0.909878i	$-0.636175\pi$
$17$	−58.1595	−0.829750	−0.414875	+	0.909878i	$0.636175\pi$
$18$	0	0
$19$	102.608	1.23895	0.619473	−	0.785018i	$-0.287346\pi$
$19$	102.608	1.23895	0.619473	+	0.785018i	$0.287346\pi$
$20$	0	0
$21$	−88.5436	−0.920085
$22$	0	0
$23$	−111.738	−1.01300	−0.506499	−	0.862241i	$-0.669060\pi$
$23$	−111.738	−1.01300	−0.506499	+	0.862241i	$0.669060\pi$
$24$	0	0
$25$	−111.429	−0.891428
$26$	0	0
$27$	−150.560	−1.07316
$28$	0	0
$29$	29.0000	0.185695
$29$	29.0000	0.185695
$30$	0	0
$31$	146.296	0.847595	0.423798	−	0.905757i	$-0.360697\pi$
$31$	146.296	0.847595	0.423798	+	0.905757i	$0.360697\pi$
$32$	0	0
$33$	63.7510	0.336292
$34$	0	0
$35$	85.3395	0.412143
$36$	0	0
$37$	77.3874	0.343849	0.171924	−	0.985110i	$-0.445002\pi$
$37$	77.3874	0.343849	0.171924	+	0.985110i	$0.445002\pi$
$38$	0	0
$39$	−336.111	−1.38002
$40$	0	0
$41$	245.696	0.935885	0.467942	−	0.883759i	$-0.344995\pi$
$41$	245.696	0.935885	0.467942	+	0.883759i	$0.344995\pi$
$42$	0	0
$43$	369.510	1.31046	0.655229	−	0.755430i	$-0.272572\pi$
$43$	369.510	1.31046	0.655229	+	0.755430i	$0.272572\pi$
$44$	0	0
$45$	45.6454	0.151209
$46$	0	0
$47$	−3.90706	−0.0121256	−0.00606279	−	0.999982i	$-0.501930\pi$
$47$	−3.90706	−0.0121256	−0.00606279	+	0.999982i	$0.501930\pi$
$48$	0	0
$49$	193.629	0.564515
$50$	0	0
$51$	−222.301	−0.610359
$52$	0	0
$53$	−355.978	−0.922592	−0.461296	−	0.887246i	$-0.652615\pi$
$53$	−355.978	−0.922592	−0.461296	+	0.887246i	$0.652615\pi$
$54$	0	0
$55$	−61.4441	−0.150639
$56$	0	0
$57$	392.196	0.911362
$58$	0	0
$59$	−223.920	−0.494100	−0.247050	−	0.969003i	$-0.579461\pi$
$59$	−223.920	−0.494100	−0.247050	+	0.969003i	$0.579461\pi$
$60$	0	0
$61$	308.765	0.648087	0.324044	−	0.946042i	$-0.394958\pi$
$61$	308.765	0.648087	0.324044	+	0.946042i	$0.394958\pi$
$62$	0	0
$63$	287.025	0.573997
$64$	0	0
$65$	323.948	0.618167
$66$	0	0
$67$	−548.837	−1.00076	−0.500382	−	0.865805i	$-0.666807\pi$
$67$	−548.837	−1.00076	−0.500382	+	0.865805i	$0.666807\pi$
$68$	0	0
$69$	−427.091	−0.745155
$70$	0	0
$71$	−121.875	−0.203717	−0.101859	−	0.994799i	$-0.532479\pi$
$71$	−121.875	−0.203717	−0.101859	+	0.994799i	$0.532479\pi$
$72$	0	0
$73$	228.283	0.366007	0.183004	−	0.983112i	$-0.441418\pi$
$73$	228.283	0.366007	0.183004	+	0.983112i	$0.441418\pi$
$74$	0	0
$75$	−425.909	−0.655729
$76$	0	0
$77$	−386.371	−0.571831
$78$	0	0
$79$	665.320	0.947524	0.473762	−	0.880653i	$-0.342896\pi$
$79$	665.320	0.947524	0.473762	+	0.880653i	$0.342896\pi$
$80$	0	0
$81$	−240.940	−0.330508
$82$	0	0
$83$	1417.96	1.87520	0.937598	−	0.347720i	$-0.113044\pi$
$83$	1417.96	1.87520	0.937598	+	0.347720i	$0.113044\pi$
$84$	0	0
$85$	214.256	0.273404
$86$	0	0
$87$	110.845	0.136596
$88$	0	0
$89$	955.874	1.13845	0.569227	−	0.822180i	$-0.307243\pi$
$89$	955.874	1.13845	0.569227	+	0.822180i	$0.307243\pi$
$90$	0	0
$91$	2037.04	2.34659
$92$	0	0
$93$	559.179	0.623486
$94$	0	0
$95$	−378.004	−0.408236
$96$	0	0
$97$	−598.996	−0.626998	−0.313499	−	0.949588i	$-0.601501\pi$
$97$	−598.996	−0.626998	−0.313499	+	0.949588i	$0.601501\pi$
$98$	0	0
$99$	−206.657	−0.209796
$100$	0	0
$101$	−373.161	−0.367633	−0.183816	−	0.982961i	$-0.558845\pi$
$101$	−373.161	−0.367633	−0.183816	+	0.982961i	$0.558845\pi$
$102$	0	0
$103$	329.720	0.315420	0.157710	−	0.987485i	$-0.449589\pi$
$103$	329.720	0.315420	0.157710	+	0.987485i	$0.449589\pi$
$104$	0	0
$105$	326.190	0.303170
$106$	0	0
$107$	−1034.22	−0.934413	−0.467207	−	0.884148i	$-0.654740\pi$
$107$	−1034.22	−0.934413	−0.467207	+	0.884148i	$0.654740\pi$
$108$	0	0
$109$	−788.911	−0.693248	−0.346624	−	0.938004i	$-0.612672\pi$
$109$	−788.911	−0.693248	−0.346624	+	0.938004i	$0.612672\pi$
$110$	0	0
$111$	295.795	0.252933
$112$	0	0
$113$	1565.48	1.30326	0.651630	−	0.758537i	$-0.274085\pi$
$113$	1565.48	1.30326	0.651630	+	0.758537i	$0.274085\pi$
$114$	0	0
$115$	411.636	0.333785
$116$	0	0
$117$	1089.55	0.860929
$118$	0	0
$119$	1347.28	1.03786
$120$	0	0
$121$	−1052.81	−0.790995
$122$	0	0
$123$	939.114	0.688431
$124$	0	0
$125$	870.990	0.623230
$126$	0	0
$127$	1733.29	1.21106	0.605529	−	0.795823i	$-0.292962\pi$
$127$	1733.29	1.21106	0.605529	+	0.795823i	$0.292962\pi$
$128$	0	0
$129$	1412.36	0.963965
$130$	0	0
$131$	1738.47	1.15947	0.579736	−	0.814805i	$-0.303156\pi$
$131$	1738.47	1.15947	0.579736	+	0.814805i	$0.303156\pi$
$132$	0	0
$133$	−2376.95	−1.54968
$134$	0	0
$135$	554.655	0.353608
$136$	0	0
$137$	2380.94	1.48480	0.742398	−	0.669959i	$-0.233688\pi$
$137$	2380.94	1.48480	0.742398	+	0.669959i	$0.233688\pi$
$138$	0	0
$139$	−2509.88	−1.53155	−0.765775	−	0.643108i	$-0.777645\pi$
$139$	−2509.88	−1.53155	−0.765775	+	0.643108i	$0.777645\pi$
$140$	0	0
$141$	−14.9338	−0.00891951
$142$	0	0
$143$	−1466.66	−0.857681
$144$	0	0
$145$	−106.834	−0.0611870
$146$	0	0
$147$	740.099	0.415254
$148$	0	0
$149$	−2367.15	−1.30151	−0.650753	−	0.759290i	$-0.725547\pi$
$149$	−2367.15	−1.30151	−0.650753	+	0.759290i	$0.725547\pi$
$150$	0	0
$151$	824.818	0.444521	0.222261	−	0.974987i	$-0.428656\pi$
$151$	824.818	0.444521	0.222261	+	0.974987i	$0.428656\pi$
$152$	0	0
$153$	720.616	0.380773
$154$	0	0
$155$	−538.945	−0.279284
$156$	0	0
$157$	−1626.71	−0.826914	−0.413457	−	0.910524i	$-0.635679\pi$
$157$	−1626.71	−0.826914	−0.413457	+	0.910524i	$0.635679\pi$
$158$	0	0
$159$	−1360.64	−0.678653
$160$	0	0
$161$	2588.43	1.26706
$162$	0	0
$163$	−1539.97	−0.739997	−0.369998	−	0.929032i	$-0.620642\pi$
$163$	−1539.97	−0.739997	−0.369998	+	0.929032i	$0.620642\pi$
$164$	0	0
$165$	−234.855	−0.110809
$166$	0	0
$167$	−1455.21	−0.674298	−0.337149	−	0.941451i	$-0.609463\pi$
$167$	−1455.21	−0.674298	−0.337149	+	0.941451i	$0.609463\pi$
$168$	0	0
$169$	5535.60	2.51962
$170$	0	0
$171$	−1271.35	−0.568554
$172$	0	0
$173$	−894.572	−0.393139	−0.196570	−	0.980490i	$-0.562980\pi$
$173$	−894.572	−0.393139	−0.196570	+	0.980490i	$0.562980\pi$
$174$	0	0
$175$	2581.27	1.11500
$176$	0	0
$177$	−855.880	−0.363457
$178$	0	0
$179$	1066.06	0.445144	0.222572	−	0.974916i	$-0.428555\pi$
$179$	1066.06	0.445144	0.222572	+	0.974916i	$0.428555\pi$
$180$	0	0
$181$	2568.24	1.05467	0.527336	−	0.849657i	$-0.323191\pi$
$181$	2568.24	1.05467	0.527336	+	0.849657i	$0.323191\pi$
$182$	0	0
$183$	1180.18	0.476729
$184$	0	0
$185$	−285.091	−0.113299
$186$	0	0
$187$	−970.035	−0.379337
$188$	0	0
$189$	3487.76	1.34231
$190$	0	0
$191$	2694.47	1.02076	0.510379	−	0.859949i	$-0.329505\pi$
$191$	2694.47	1.02076	0.510379	+	0.859949i	$0.329505\pi$
$192$	0	0
$193$	4017.01	1.49819	0.749096	−	0.662462i	$-0.230488\pi$
$193$	4017.01	1.49819	0.749096	+	0.662462i	$0.230488\pi$
$194$	0	0
$195$	1238.21	0.454720
$196$	0	0
$197$	2152.43	0.778449	0.389225	−	0.921143i	$-0.372743\pi$
$197$	2152.43	0.778449	0.389225	+	0.921143i	$0.372743\pi$
$198$	0	0
$199$	−4149.11	−1.47800	−0.739002	−	0.673703i	$-0.764703\pi$
$199$	−4149.11	−1.47800	−0.739002	+	0.673703i	$0.764703\pi$
$200$	0	0
$201$	−2097.80	−0.736155
$202$	0	0
$203$	−671.792	−0.232269
$204$	0	0
$205$	−905.131	−0.308376
$206$	0	0
$207$	1384.47	0.464866
$208$	0	0
$209$	1711.39	0.566410
$210$	0	0
$211$	−2936.09	−0.957956	−0.478978	−	0.877827i	$-0.658993\pi$
$211$	−2936.09	−0.957956	−0.478978	+	0.877827i	$0.658993\pi$
$212$	0	0
$213$	−465.839	−0.149853
$214$	0	0
$215$	−1361.25	−0.431799
$216$	0	0
$217$	−3388.97	−1.06018
$218$	0	0
$219$	872.557	0.269233
$220$	0	0
$221$	5114.26	1.55666
$222$	0	0
$223$	−1093.98	−0.328511	−0.164256	−	0.986418i	$-0.552522\pi$
$223$	−1093.98	−0.328511	−0.164256	+	0.986418i	$0.552522\pi$
$224$	0	0
$225$	1380.64	0.409078
$226$	0	0
$227$	−3131.85	−0.915718	−0.457859	−	0.889025i	$-0.651383\pi$
$227$	−3131.85	−0.915718	−0.457859	+	0.889025i	$0.651383\pi$
$228$	0	0
$229$	3047.47	0.879401	0.439700	−	0.898145i	$-0.355085\pi$
$229$	3047.47	0.879401	0.439700	+	0.898145i	$0.355085\pi$
$230$	0	0
$231$	−1476.81	−0.420636
$232$	0	0
$233$	−5271.53	−1.48219	−0.741093	−	0.671402i	$-0.765692\pi$
$233$	−5271.53	−1.48219	−0.741093	+	0.671402i	$0.765692\pi$
$234$	0	0
$235$	14.3934	0.00399541
$236$	0	0
$237$	2543.03	0.696993
$238$	0	0
$239$	4925.29	1.33302	0.666508	−	0.745498i	$-0.267788\pi$
$239$	4925.29	1.33302	0.666508	+	0.745498i	$0.267788\pi$
$240$	0	0
$241$	5626.99	1.50401	0.752004	−	0.659158i	$-0.229087\pi$
$241$	5626.99	1.50401	0.752004	+	0.659158i	$0.229087\pi$
$242$	0	0
$243$	3144.19	0.830040
$244$	0	0
$245$	−713.318	−0.186009
$246$	0	0
$247$	−9022.89	−2.32434
$248$	0	0
$249$	5419.81	1.37938
$250$	0	0
$251$	1130.42	0.284270	0.142135	−	0.989847i	$-0.454603\pi$
$251$	1130.42	0.284270	0.142135	+	0.989847i	$0.454603\pi$
$252$	0	0
$253$	−1863.66	−0.463112
$254$	0	0
$255$	818.943	0.201115
$256$	0	0
$257$	2833.90	0.687836	0.343918	−	0.939000i	$-0.388246\pi$
$257$	2833.90	0.687836	0.343918	+	0.939000i	$0.388246\pi$
$258$	0	0
$259$	−1792.70	−0.430088
$260$	0	0
$261$	−359.320	−0.0852159
$262$	0	0
$263$	−2976.83	−0.697943	−0.348971	−	0.937133i	$-0.613469\pi$
$263$	−2976.83	−0.697943	−0.348971	+	0.937133i	$0.613469\pi$
$264$	0	0
$265$	1311.41	0.303996
$266$	0	0
$267$	3653.60	0.837440
$268$	0	0
$269$	−3207.93	−0.727105	−0.363552	−	0.931574i	$-0.618436\pi$
$269$	−3207.93	−0.727105	−0.363552	+	0.931574i	$0.618436\pi$
$270$	0	0
$271$	−8191.64	−1.83619	−0.918094	−	0.396363i	$-0.870272\pi$
$271$	−8191.64	−1.83619	−0.918094	+	0.396363i	$0.870272\pi$
$272$	0	0
$273$	7786.09	1.72614
$274$	0	0
$275$	−1858.50	−0.407535
$276$	0	0
$277$	1005.50	0.218104	0.109052	−	0.994036i	$-0.465218\pi$
$277$	1005.50	0.218104	0.109052	+	0.994036i	$0.465218\pi$
$278$	0	0
$279$	−1812.65	−0.388963
$280$	0	0
$281$	−3267.81	−0.693741	−0.346870	−	0.937913i	$-0.612756\pi$
$281$	−3267.81	−0.693741	−0.346870	+	0.937913i	$0.612756\pi$
$282$	0	0
$283$	−9245.89	−1.94209	−0.971044	−	0.238900i	$-0.923213\pi$
$283$	−9245.89	−1.94209	−0.971044	+	0.238900i	$0.923213\pi$
$284$	0	0
$285$	−1444.83	−0.300296
$286$	0	0
$287$	−5691.61	−1.17061
$288$	0	0
$289$	−1530.48	−0.311515
$290$	0	0
$291$	−2289.52	−0.461216
$292$	0	0
$293$	5610.98	1.11876	0.559381	−	0.828911i	$-0.311039\pi$
$293$	5610.98	1.11876	0.559381	+	0.828911i	$0.311039\pi$
$294$	0	0
$295$	824.909	0.162807
$296$	0	0
$297$	−2511.17	−0.490617
$298$	0	0
$299$	9825.68	1.90045
$300$	0	0
$301$	−8559.79	−1.63913
$302$	0	0
$303$	−1426.32	−0.270429
$304$	0	0
$305$	−1137.47	−0.213546
$306$	0	0
$307$	−2954.97	−0.549346	−0.274673	−	0.961538i	$-0.588570\pi$
$307$	−2954.97	−0.549346	−0.274673	+	0.961538i	$0.588570\pi$
$308$	0	0
$309$	1260.28	0.232021
$310$	0	0
$311$	−3138.95	−0.572327	−0.286163	−	0.958181i	$-0.592380\pi$
$311$	−3138.95	−0.572327	−0.286163	+	0.958181i	$0.592380\pi$
$312$	0	0
$313$	−7784.41	−1.40575	−0.702876	−	0.711312i	$-0.748101\pi$
$313$	−7784.41	−1.40575	−0.702876	+	0.711312i	$0.748101\pi$
$314$	0	0
$315$	−1057.39	−0.189133
$316$	0	0
$317$	10623.9	1.88233	0.941165	−	0.337947i	$-0.109732\pi$
$317$	10623.9	1.88233	0.941165	+	0.337947i	$0.109732\pi$
$318$	0	0
$319$	483.688	0.0848944
$320$	0	0
$321$	−3953.07	−0.687349
$322$	0	0
$323$	−5967.65	−1.02802
$324$	0	0
$325$	9798.49	1.67238
$326$	0	0
$327$	−3015.42	−0.509949
$328$	0	0
$329$	90.5079	0.0151668
$330$	0	0
$331$	6053.90	1.00529	0.502647	−	0.864491i	$-0.332359\pi$
$331$	6053.90	1.00529	0.502647	+	0.864491i	$0.332359\pi$
$332$	0	0
$333$	−958.856	−0.157793
$334$	0	0
$335$	2021.89	0.329754
$336$	0	0
$337$	2776.81	0.448850	0.224425	−	0.974491i	$-0.427950\pi$
$337$	2776.81	0.448850	0.224425	+	0.974491i	$0.427950\pi$
$338$	0	0
$339$	5983.69	0.958670
$340$	0	0
$341$	2440.05	0.387495
$342$	0	0
$343$	3460.22	0.544707
$344$	0	0
$345$	1573.38	0.245530
$346$	0	0
$347$	8090.38	1.25163	0.625814	−	0.779973i	$-0.284767\pi$
$347$	8090.38	1.25163	0.625814	+	0.779973i	$0.284767\pi$
$348$	0	0
$349$	4261.31	0.653589	0.326795	−	0.945095i	$-0.394031\pi$
$349$	4261.31	0.653589	0.326795	+	0.945095i	$0.394031\pi$
$350$	0	0
$351$	13239.5	2.01332
$352$	0	0
$353$	−6633.39	−1.00017	−0.500085	−	0.865977i	$-0.666698\pi$
$353$	−6633.39	−1.00017	−0.500085	+	0.865977i	$0.666698\pi$
$354$	0	0
$355$	448.982	0.0671253
$356$	0	0
$357$	5149.65	0.763440
$358$	0	0
$359$	6341.48	0.932285	0.466143	−	0.884710i	$-0.345643\pi$
$359$	6341.48	0.932285	0.466143	+	0.884710i	$0.345643\pi$
$360$	0	0
$361$	3669.49	0.534988
$362$	0	0
$363$	−4024.13	−0.581851
$364$	0	0
$365$	−840.983	−0.120600
$366$	0	0
$367$	6111.88	0.869312	0.434656	−	0.900597i	$-0.356870\pi$
$367$	6111.88	0.869312	0.434656	+	0.900597i	$0.356870\pi$
$368$	0	0
$369$	−3044.26	−0.429479
$370$	0	0
$371$	8246.33	1.15398
$372$	0	0
$373$	1987.66	0.275917	0.137958	−	0.990438i	$-0.455946\pi$
$373$	1987.66	0.275917	0.137958	+	0.990438i	$0.455946\pi$
$374$	0	0
$375$	3329.15	0.458444
$376$	0	0
$377$	−2550.12	−0.348376
$378$	0	0
$379$	2188.51	0.296612	0.148306	−	0.988941i	$-0.452618\pi$
$379$	2188.51	0.296612	0.148306	+	0.988941i	$0.452618\pi$
$380$	0	0
$381$	6625.07	0.890847
$382$	0	0
$383$	9799.64	1.30741	0.653705	−	0.756749i	$-0.273214\pi$
$383$	9799.64	1.30741	0.653705	+	0.756749i	$0.273214\pi$
$384$	0	0
$385$	1423.37	0.188420
$386$	0	0
$387$	−4578.35	−0.601371
$388$	0	0
$389$	−8102.72	−1.05610	−0.528052	−	0.849212i	$-0.677077\pi$
$389$	−8102.72	−1.05610	−0.528052	+	0.849212i	$0.677077\pi$
$390$	0	0
$391$	6498.61	0.840534
$392$	0	0
$393$	6644.88	0.852900
$394$	0	0
$395$	−2451.00	−0.312211
$396$	0	0
$397$	13884.0	1.75521	0.877604	−	0.479386i	$-0.159141\pi$
$397$	13884.0	1.75521	0.877604	+	0.479386i	$0.159141\pi$
$398$	0	0
$399$	−9085.31	−1.13994
$400$	0	0
$401$	7173.11	0.893287	0.446643	−	0.894712i	$-0.352619\pi$
$401$	7173.11	0.893287	0.446643	+	0.894712i	$0.352619\pi$
$402$	0	0
$403$	−12864.5	−1.59014
$404$	0	0
$405$	887.611	0.108903
$406$	0	0
$407$	1290.74	0.157198
$408$	0	0
$409$	4796.88	0.579928	0.289964	−	0.957038i	$-0.406357\pi$
$409$	4796.88	0.579928	0.289964	+	0.957038i	$0.406357\pi$
$410$	0	0
$411$	9100.56	1.09221
$412$	0	0
$413$	5187.16	0.618023
$414$	0	0
$415$	−5223.69	−0.617881
$416$	0	0
$417$	−9593.42	−1.12660
$418$	0	0
$419$	−8774.07	−1.02301	−0.511505	−	0.859280i	$-0.670912\pi$
$419$	−8774.07	−1.02301	−0.511505	+	0.859280i	$0.670912\pi$
$420$	0	0
$421$	5603.65	0.648706	0.324353	−	0.945936i	$-0.394853\pi$
$421$	5603.65	0.648706	0.324353	+	0.945936i	$0.394853\pi$
$422$	0	0
$423$	48.4098	0.00556445
$424$	0	0
$425$	6480.63	0.739662
$426$	0	0
$427$	−7152.62	−0.810631
$428$	0	0
$429$	−5605.96	−0.630905
$430$	0	0
$431$	8895.48	0.994153	0.497077	−	0.867707i	$-0.334407\pi$
$431$	8895.48	0.994153	0.497077	+	0.867707i	$0.334407\pi$
$432$	0	0
$433$	−5441.65	−0.603947	−0.301973	−	0.953316i	$-0.597645\pi$
$433$	−5441.65	−0.603947	−0.301973	+	0.953316i	$0.597645\pi$
$434$	0	0
$435$	−408.349	−0.0450088
$436$	0	0
$437$	−11465.2	−1.25505
$438$	0	0
$439$	11470.9	1.24710	0.623550	−	0.781783i	$-0.285690\pi$
$439$	11470.9	1.24710	0.623550	+	0.781783i	$0.285690\pi$
$440$	0	0
$441$	−2399.13	−0.259057
$442$	0	0
$443$	−1078.07	−0.115623	−0.0578113	−	0.998328i	$-0.518412\pi$
$443$	−1078.07	−0.115623	−0.0578113	+	0.998328i	$0.518412\pi$
$444$	0	0
$445$	−3521.39	−0.375123
$446$	0	0
$447$	−9047.85	−0.957380
$448$	0	0
$449$	2374.52	0.249578	0.124789	−	0.992183i	$-0.460175\pi$
$449$	2374.52	0.249578	0.124789	+	0.992183i	$0.460175\pi$
$450$	0	0
$451$	4097.94	0.427859
$452$	0	0
$453$	3152.67	0.326987
$454$	0	0
$455$	−7504.35	−0.773207
$456$	0	0
$457$	−10845.0	−1.11008	−0.555040	−	0.831823i	$-0.687297\pi$
$457$	−10845.0	−1.11008	−0.555040	+	0.831823i	$0.687297\pi$
$458$	0	0
$459$	8756.49	0.890453
$460$	0	0
$461$	12520.9	1.26498	0.632491	−	0.774568i	$-0.282033\pi$
$461$	12520.9	1.26498	0.632491	+	0.774568i	$0.282033\pi$
$462$	0	0
$463$	−12163.8	−1.22095	−0.610474	−	0.792036i	$-0.709021\pi$
$463$	−12163.8	−1.22095	−0.610474	+	0.792036i	$0.709021\pi$
$464$	0	0
$465$	−2059.99	−0.205440
$466$	0	0
$467$	−13079.3	−1.29601	−0.648005	−	0.761636i	$-0.724396\pi$
$467$	−13079.3	−1.29601	−0.648005	+	0.761636i	$0.724396\pi$
$468$	0	0
$469$	12713.9	1.25176
$470$	0	0
$471$	−6217.70	−0.608273
$472$	0	0
$473$	6163.01	0.599103
$474$	0	0
$475$	−11433.5	−1.10443
$476$	0	0
$477$	4410.69	0.423379
$478$	0	0
$479$	10545.6	1.00593	0.502963	−	0.864308i	$-0.332243\pi$
$479$	10545.6	1.00593	0.502963	+	0.864308i	$0.332243\pi$
$480$	0	0
$481$	−6805.07	−0.645083
$482$	0	0
$483$	9893.66	0.932044
$484$	0	0
$485$	2206.67	0.206597
$486$	0	0
$487$	−4103.54	−0.381826	−0.190913	−	0.981607i	$-0.561145\pi$
$487$	−4103.54	−0.381826	−0.190913	+	0.981607i	$0.561145\pi$
$488$	0	0
$489$	−5886.15	−0.544337
$490$	0	0
$491$	12457.3	1.14499	0.572496	−	0.819908i	$-0.305975\pi$
$491$	12457.3	1.14499	0.572496	+	0.819908i	$0.305975\pi$
$492$	0	0
$493$	−1686.62	−0.154081
$494$	0	0
$495$	761.314	0.0691283
$496$	0	0
$497$	2823.27	0.254811
$498$	0	0
$499$	−7508.63	−0.673612	−0.336806	−	0.941574i	$-0.609347\pi$
$499$	−7508.63	−0.673612	−0.336806	+	0.941574i	$0.609347\pi$
$500$	0	0
$501$	−5562.20	−0.496010
$502$	0	0
$503$	11666.7	1.03418	0.517088	−	0.855932i	$-0.327016\pi$
$503$	11666.7	1.03418	0.517088	+	0.855932i	$0.327016\pi$
$504$	0	0
$505$	1374.71	0.121136
$506$	0	0
$507$	21158.5	1.85341
$508$	0	0
$509$	6641.16	0.578319	0.289159	−	0.957281i	$-0.406624\pi$
$509$	6641.16	0.578319	0.289159	+	0.957281i	$0.406624\pi$
$510$	0	0
$511$	−5288.24	−0.457804
$512$	0	0
$513$	−15448.7	−1.32959
$514$	0	0
$515$	−1214.67	−0.103932
$516$	0	0
$517$	−65.1654	−0.00554346
$518$	0	0
$519$	−3419.29	−0.289191
$520$	0	0
$521$	−3842.82	−0.323142	−0.161571	−	0.986861i	$-0.551656\pi$
$521$	−3842.82	−0.323142	−0.161571	+	0.986861i	$0.551656\pi$
$522$	0	0
$523$	−13615.6	−1.13838	−0.569188	−	0.822208i	$-0.692742\pi$
$523$	−13615.6	−1.13838	−0.569188	+	0.822208i	$0.692742\pi$
$524$	0	0
$525$	9866.28	0.820190
$526$	0	0
$527$	−8508.47	−0.703292
$528$	0	0
$529$	318.329	0.0261633
$530$	0	0
$531$	2774.45	0.226743
$532$	0	0
$533$	−21605.3	−1.75578
$534$	0	0
$535$	3810.03	0.307891
$536$	0	0
$537$	4074.74	0.327445
$538$	0	0
$539$	3229.51	0.258080
$540$	0	0
$541$	22963.9	1.82494	0.912472	−	0.409140i	$-0.134171\pi$
$541$	22963.9	1.82494	0.912472	+	0.409140i	$0.134171\pi$
$542$	0	0
$543$	9816.47	0.775810
$544$	0	0
$545$	2906.31	0.228427
$546$	0	0
$547$	7229.04	0.565066	0.282533	−	0.959258i	$-0.408825\pi$
$547$	7229.04	0.565066	0.282533	+	0.959258i	$0.408825\pi$
$548$	0	0
$549$	−3825.71	−0.297408
$550$	0	0
$551$	2975.64	0.230067
$552$	0	0
$553$	−15412.3	−1.18517
$554$	0	0
$555$	−1089.69	−0.0833420
$556$	0	0
$557$	−12409.3	−0.943981	−0.471991	−	0.881604i	$-0.656464\pi$
$557$	−12409.3	−0.943981	−0.471991	+	0.881604i	$0.656464\pi$
$558$	0	0
$559$	−32492.9	−2.45850
$560$	0	0
$561$	−3707.73	−0.279038
$562$	0	0
$563$	−21172.6	−1.58493	−0.792467	−	0.609915i	$-0.791204\pi$
$563$	−21172.6	−1.58493	−0.792467	+	0.609915i	$0.791204\pi$
$564$	0	0
$565$	−5767.16	−0.429427
$566$	0	0
$567$	5581.44	0.413401
$568$	0	0
$569$	−5160.24	−0.380191	−0.190096	−	0.981766i	$-0.560880\pi$
$569$	−5160.24	−0.380191	−0.190096	+	0.981766i	$0.560880\pi$
$570$	0	0
$571$	20730.0	1.51931	0.759653	−	0.650329i	$-0.225369\pi$
$571$	20730.0	1.51931	0.759653	+	0.650329i	$0.225369\pi$
$572$	0	0
$573$	10299.0	0.750864
$574$	0	0
$575$	12450.8	0.903014
$576$	0	0
$577$	16086.6	1.16065	0.580325	−	0.814385i	$-0.302926\pi$
$577$	16086.6	1.16065	0.580325	+	0.814385i	$0.302926\pi$
$578$	0	0
$579$	15354.1	1.10206
$580$	0	0
$581$	−32847.4	−2.34551
$582$	0	0
$583$	−5937.32	−0.421782
$584$	0	0
$585$	−4013.83	−0.283678
$586$	0	0
$587$	−23506.7	−1.65285	−0.826426	−	0.563046i	$-0.809629\pi$
$587$	−23506.7	−1.65285	−0.826426	+	0.563046i	$0.809629\pi$
$588$	0	0
$589$	15011.1	1.05012
$590$	0	0
$591$	8227.15	0.572622
$592$	0	0
$593$	−190.876	−0.0132181	−0.00660904	−	0.999978i	$-0.502104\pi$
$593$	−190.876	−0.0132181	−0.00660904	+	0.999978i	$0.502104\pi$
$594$	0	0
$595$	−4963.30	−0.341976
$596$	0	0
$597$	−15859.0	−1.08721
$598$	0	0
$599$	−8606.05	−0.587034	−0.293517	−	0.955954i	$-0.594826\pi$
$599$	−8606.05	−0.587034	−0.293517	+	0.955954i	$0.594826\pi$
$600$	0	0
$601$	−694.830	−0.0471592	−0.0235796	−	0.999722i	$-0.507506\pi$
$601$	−694.830	−0.0471592	−0.0235796	+	0.999722i	$0.507506\pi$
$602$	0	0
$603$	6800.28	0.459252
$604$	0	0
$605$	3878.51	0.260635
$606$	0	0
$607$	21056.6	1.40801	0.704005	−	0.710195i	$-0.251393\pi$
$607$	21056.6	1.40801	0.704005	+	0.710195i	$0.251393\pi$
$608$	0	0
$609$	−2567.76	−0.170856
$610$	0	0
$611$	343.568	0.0227484
$612$	0	0
$613$	21876.7	1.44142	0.720712	−	0.693234i	$-0.243815\pi$
$613$	21876.7	1.44142	0.720712	+	0.693234i	$0.243815\pi$
$614$	0	0
$615$	−3459.64	−0.226840
$616$	0	0
$617$	317.097	0.0206902	0.0103451	−	0.999946i	$-0.496707\pi$
$617$	317.097	0.0206902	0.0103451	+	0.999946i	$0.496707\pi$
$618$	0	0
$619$	21304.3	1.38334	0.691672	−	0.722212i	$-0.256874\pi$
$619$	21304.3	1.38334	0.691672	+	0.722212i	$0.256874\pi$
$620$	0	0
$621$	16823.2	1.08711
$622$	0	0
$623$	−22143.1	−1.42399
$624$	0	0
$625$	10719.9	0.686073
$626$	0	0
$627$	6541.39	0.416647
$628$	0	0
$629$	−4500.81	−0.285308
$630$	0	0
$631$	3859.09	0.243468	0.121734	−	0.992563i	$-0.461155\pi$
$631$	3859.09	0.243468	0.121734	+	0.992563i	$0.461155\pi$
$632$	0	0
$633$	−11222.5	−0.704667
$634$	0	0
$635$	−6385.34	−0.399046
$636$	0	0
$637$	−17026.8	−1.05907
$638$	0	0
$639$	1510.08	0.0934862
$640$	0	0
$641$	8186.84	0.504463	0.252231	−	0.967667i	$-0.418836\pi$
$641$	8186.84	0.504463	0.252231	+	0.967667i	$0.418836\pi$
$642$	0	0
$643$	5319.43	0.326248	0.163124	−	0.986606i	$-0.447843\pi$
$643$	5319.43	0.326248	0.163124	+	0.986606i	$0.447843\pi$
$644$	0	0
$645$	−5203.06	−0.317629
$646$	0	0
$647$	8274.24	0.502773	0.251386	−	0.967887i	$-0.419114\pi$
$647$	8274.24	0.502773	0.251386	+	0.967887i	$0.419114\pi$
$648$	0	0
$649$	−3734.74	−0.225888
$650$	0	0
$651$	−12953.5	−0.779860
$652$	0	0
$653$	−14507.8	−0.869422	−0.434711	−	0.900570i	$-0.643149\pi$
$653$	−14507.8	−0.869422	−0.434711	+	0.900570i	$0.643149\pi$
$654$	0	0
$655$	−6404.43	−0.382048
$656$	0	0
$657$	−2828.51	−0.167961
$658$	0	0
$659$	−12081.9	−0.714177	−0.357089	−	0.934071i	$-0.616231\pi$
$659$	−12081.9	−0.714177	−0.357089	+	0.934071i	$0.616231\pi$
$660$	0	0
$661$	−15937.2	−0.937798	−0.468899	−	0.883252i	$-0.655349\pi$
$661$	−15937.2	−0.937798	−0.468899	+	0.883252i	$0.655349\pi$
$662$	0	0
$663$	19548.0	1.14507
$664$	0	0
$665$	8756.55	0.510623
$666$	0	0
$667$	−3240.40	−0.188109
$668$	0	0
$669$	−4181.46	−0.241651
$670$	0	0
$671$	5149.86	0.296286
$672$	0	0
$673$	17165.6	0.983190	0.491595	−	0.870824i	$-0.336414\pi$
$673$	17165.6	0.983190	0.491595	+	0.870824i	$0.336414\pi$
$674$	0	0
$675$	16776.7	0.956644
$676$	0	0
$677$	−19033.0	−1.08050	−0.540248	−	0.841506i	$-0.681670\pi$
$677$	−19033.0	−1.08050	−0.540248	+	0.841506i	$0.681670\pi$
$678$	0	0
$679$	13875.9	0.784253
$680$	0	0
$681$	−11970.7	−0.673596
$682$	0	0
$683$	−5918.45	−0.331571	−0.165786	−	0.986162i	$-0.553016\pi$
$683$	−5918.45	−0.331571	−0.165786	+	0.986162i	$0.553016\pi$
$684$	0	0
$685$	−8771.24	−0.489244
$686$	0	0
$687$	11648.2	0.646882
$688$	0	0
$689$	31303.0	1.73084
$690$	0	0
$691$	18611.5	1.02462	0.512311	−	0.858800i	$-0.328790\pi$
$691$	18611.5	1.02462	0.512311	+	0.858800i	$0.328790\pi$
$692$	0	0
$693$	4787.26	0.262414
$694$	0	0
$695$	9246.28	0.504649
$696$	0	0
$697$	−14289.6	−0.776550
$698$	0	0
$699$	−20149.1	−1.09029
$700$	0	0
$701$	−32563.7	−1.75451	−0.877257	−	0.480022i	$-0.840629\pi$
$701$	−32563.7	−1.75451	−0.877257	+	0.480022i	$0.840629\pi$
$702$	0	0
$703$	7940.60	0.426010
$704$	0	0
$705$	55.0152	0.00293900
$706$	0	0
$707$	8644.37	0.459837
$708$	0	0
$709$	18550.0	0.982595	0.491297	−	0.870992i	$-0.336523\pi$
$709$	18550.0	0.982595	0.491297	+	0.870992i	$0.336523\pi$
$710$	0	0
$711$	−8243.55	−0.434820
$712$	0	0
$713$	−16346.7	−0.858611
$714$	0	0
$715$	5403.10	0.282608
$716$	0	0
$717$	18825.7	0.980559
$718$	0	0
$719$	−29319.9	−1.52079	−0.760395	−	0.649461i	$-0.774994\pi$
$719$	−29319.9	−1.52079	−0.760395	+	0.649461i	$0.774994\pi$
$720$	0	0
$721$	−7638.05	−0.394530
$722$	0	0
$723$	21507.8	1.10634
$724$	0	0
$725$	−3231.43	−0.165534
$726$	0	0
$727$	−24587.3	−1.25432	−0.627161	−	0.778890i	$-0.715783\pi$
$727$	−24587.3	−1.25432	−0.627161	+	0.778890i	$0.715783\pi$
$728$	0	0
$729$	18523.3	0.941080
$730$	0	0
$731$	−21490.5	−1.08735
$732$	0	0
$733$	−3703.60	−0.186624	−0.0933122	−	0.995637i	$-0.529745\pi$
$733$	−3703.60	−0.186624	−0.0933122	+	0.995637i	$0.529745\pi$
$734$	0	0
$735$	−2726.48	−0.136827
$736$	0	0
$737$	−9153.99	−0.457519
$738$	0	0
$739$	−4504.54	−0.224225	−0.112112	−	0.993696i	$-0.535762\pi$
$739$	−4504.54	−0.224225	−0.112112	+	0.993696i	$0.535762\pi$
$740$	0	0
$741$	−34487.8	−1.70977
$742$	0	0
$743$	−8830.39	−0.436010	−0.218005	−	0.975948i	$-0.569955\pi$
$743$	−8830.39	−0.436010	−0.218005	+	0.975948i	$0.569955\pi$
$744$	0	0
$745$	8720.45	0.428849
$746$	0	0
$747$	−17569.0	−0.860531
$748$	0	0
$749$	23958.1	1.16877
$750$	0	0
$751$	130.977	0.00636405	0.00318202	−	0.999995i	$-0.498987\pi$
$751$	130.977	0.00636405	0.00318202	+	0.999995i	$0.498987\pi$
$752$	0	0
$753$	4320.77	0.209107
$754$	0	0
$755$	−3038.58	−0.146471
$756$	0	0
$757$	−6200.63	−0.297709	−0.148855	−	0.988859i	$-0.547559\pi$
$757$	−6200.63	−0.297709	−0.148855	+	0.988859i	$0.547559\pi$
$758$	0	0
$759$	−7123.40	−0.340663
$760$	0	0
$761$	−24926.0	−1.18734	−0.593672	−	0.804707i	$-0.702322\pi$
$761$	−24926.0	−1.18734	−0.593672	+	0.804707i	$0.702322\pi$
$762$	0	0
$763$	18275.3	0.867118
$764$	0	0
$765$	−2654.71	−0.125466
$766$	0	0
$767$	19690.5	0.926963
$768$	0	0
$769$	28591.2	1.34074	0.670368	−	0.742029i	$-0.266136\pi$
$769$	28591.2	1.34074	0.670368	+	0.742029i	$0.266136\pi$
$770$	0	0
$771$	10831.9	0.505968
$772$	0	0
$773$	−7798.50	−0.362862	−0.181431	−	0.983404i	$-0.558073\pi$
$773$	−7798.50	−0.362862	−0.181431	+	0.983404i	$0.558073\pi$
$774$	0	0
$775$	−16301.5	−0.755570
$776$	0	0
$777$	−6852.16	−0.316370
$778$	0	0
$779$	25210.5	1.15951
$780$	0	0
$781$	−2032.74	−0.0931335
$782$	0	0
$783$	−4366.24	−0.199281
$784$	0	0
$785$	5992.71	0.272470
$786$	0	0
$787$	−18325.2	−0.830014	−0.415007	−	0.909818i	$-0.636221\pi$
$787$	−18325.2	−0.830014	−0.415007	+	0.909818i	$0.636221\pi$
$788$	0	0
$789$	−11378.2	−0.513402
$790$	0	0
$791$	−36264.8	−1.63013
$792$	0	0
$793$	−27151.3	−1.21585
$794$	0	0
$795$	5012.53	0.223618
$796$	0	0
$797$	−39617.3	−1.76075	−0.880374	−	0.474280i	$-0.842709\pi$
$797$	−39617.3	−1.76075	−0.880374	+	0.474280i	$0.842709\pi$
$798$	0	0
$799$	227.232	0.0100612
$800$	0	0
$801$	−11843.6	−0.522438
$802$	0	0
$803$	3807.51	0.167328
$804$	0	0
$805$	−9535.65	−0.417500
$806$	0	0
$807$	−12261.6	−0.534854
$808$	0	0
$809$	5282.05	0.229551	0.114776	−	0.993391i	$-0.463385\pi$
$809$	5282.05	0.229551	0.114776	+	0.993391i	$0.463385\pi$
$810$	0	0
$811$	20291.5	0.878582	0.439291	−	0.898345i	$-0.355230\pi$
$811$	20291.5	0.878582	0.439291	+	0.898345i	$0.355230\pi$
$812$	0	0
$813$	−31310.6	−1.35069
$814$	0	0
$815$	5673.15	0.243831
$816$	0	0
$817$	37914.8	1.62359
$818$	0	0
$819$	−25239.6	−1.07685
$820$	0	0
$821$	−21042.5	−0.894506	−0.447253	−	0.894407i	$-0.647598\pi$
$821$	−21042.5	−0.894506	−0.447253	+	0.894407i	$0.647598\pi$
$822$	0	0
$823$	31069.4	1.31593	0.657966	−	0.753048i	$-0.271417\pi$
$823$	31069.4	1.31593	0.657966	+	0.753048i	$0.271417\pi$
$824$	0	0
$825$	−7103.68	−0.299780
$826$	0	0
$827$	33573.5	1.41169	0.705843	−	0.708368i	$-0.250568\pi$
$827$	33573.5	1.41169	0.705843	+	0.708368i	$0.250568\pi$
$828$	0	0
$829$	1879.16	0.0787285	0.0393643	−	0.999225i	$-0.487467\pi$
$829$	1879.16	0.0787285	0.0393643	+	0.999225i	$0.487467\pi$
$830$	0	0
$831$	3843.30	0.160436
$832$	0	0
$833$	−11261.3	−0.468406
$834$	0	0
$835$	5360.93	0.222183
$836$	0	0
$837$	−22026.3	−0.909604
$838$	0	0
$839$	−36762.9	−1.51275	−0.756375	−	0.654139i	$-0.773031\pi$
$839$	−36762.9	−1.51275	−0.756375	+	0.654139i	$0.773031\pi$
$840$	0	0
$841$	841.000	0.0344828
$842$	0	0
$843$	−12490.4	−0.510311
$844$	0	0
$845$	−20392.8	−0.830219
$846$	0	0
$847$	24388.7	0.989381
$848$	0	0
$849$	−35340.2	−1.42859
$850$	0	0
$851$	−8647.09	−0.348318
$852$	0	0
$853$	−18398.0	−0.738495	−0.369248	−	0.929331i	$-0.620385\pi$
$853$	−18398.0	−0.738495	−0.369248	+	0.929331i	$0.620385\pi$
$854$	0	0
$855$	4683.60	0.187340
$856$	0	0
$857$	8340.82	0.332458	0.166229	−	0.986087i	$-0.446841\pi$
$857$	8340.82	0.332458	0.166229	+	0.986087i	$0.446841\pi$
$858$	0	0
$859$	−32630.4	−1.29608	−0.648041	−	0.761606i	$-0.724411\pi$
$859$	−32630.4	−1.29608	−0.648041	+	0.761606i	$0.724411\pi$
$860$	0	0
$861$	−21754.8	−0.861094
$862$	0	0
$863$	47752.0	1.88354	0.941772	−	0.336252i	$-0.109159\pi$
$863$	47752.0	1.88354	0.941772	+	0.336252i	$0.109159\pi$
$864$	0	0
$865$	3295.56	0.129540
$866$	0	0
$867$	−5849.87	−0.229149
$868$	0	0
$869$	11096.8	0.433180
$870$	0	0
$871$	48262.1	1.87750
$872$	0	0
$873$	7421.77	0.287730
$874$	0	0
$875$	−20176.7	−0.779539
$876$	0	0
$877$	−26821.6	−1.03273	−0.516363	−	0.856370i	$-0.672715\pi$
$877$	−26821.6	−1.03273	−0.516363	+	0.856370i	$0.672715\pi$
$878$	0	0
$879$	21446.6	0.822954
$880$	0	0
$881$	−20894.3	−0.799032	−0.399516	−	0.916726i	$-0.630822\pi$
$881$	−20894.3	−0.799032	−0.399516	+	0.916726i	$0.630822\pi$
$882$	0	0
$883$	5901.12	0.224902	0.112451	−	0.993657i	$-0.464130\pi$
$883$	5901.12	0.224902	0.112451	+	0.993657i	$0.464130\pi$
$884$	0	0
$885$	3153.02	0.119760
$886$	0	0
$887$	9152.53	0.346462	0.173231	−	0.984881i	$-0.444579\pi$
$887$	9152.53	0.346462	0.173231	+	0.984881i	$0.444579\pi$
$888$	0	0
$889$	−40152.0	−1.51480
$890$	0	0
$891$	−4018.61	−0.151098
$892$	0	0
$893$	−400.897	−0.0150230
$894$	0	0
$895$	−3927.29	−0.146676
$896$	0	0
$897$	37556.3	1.39796
$898$	0	0
$899$	4242.57	0.157394
$900$	0	0
$901$	20703.5	0.765521
$902$	0	0
$903$	−32717.7	−1.20573
$904$	0	0
$905$	−9461.25	−0.347517
$906$	0	0
$907$	−35746.8	−1.30866	−0.654329	−	0.756210i	$-0.727049\pi$
$907$	−35746.8	−1.30866	−0.654329	+	0.756210i	$0.727049\pi$
$908$	0	0
$909$	4623.60	0.168707
$910$	0	0
$911$	−30334.5	−1.10321	−0.551607	−	0.834104i	$-0.685985\pi$
$911$	−30334.5	−1.10321	−0.551607	+	0.834104i	$0.685985\pi$
$912$	0	0
$913$	23650.0	0.857284
$914$	0	0
$915$	−4347.72	−0.157083
$916$	0	0
$917$	−40272.1	−1.45027
$918$	0	0
$919$	8015.10	0.287697	0.143849	−	0.989600i	$-0.454052\pi$
$919$	8015.10	0.287697	0.143849	+	0.989600i	$0.454052\pi$
$920$	0	0
$921$	−11294.7	−0.404096
$922$	0	0
$923$	10717.1	0.382187
$924$	0	0
$925$	−8623.16	−0.306517
$926$	0	0
$927$	−4085.35	−0.144747
$928$	0	0
$929$	17041.8	0.601855	0.300928	−	0.953647i	$-0.402704\pi$
$929$	17041.8	0.601855	0.300928	+	0.953647i	$0.402704\pi$
$930$	0	0
$931$	19867.9	0.699404
$932$	0	0
$933$	−11997.9	−0.421000
$934$	0	0
$935$	3573.56	0.124992
$936$	0	0
$937$	−24408.7	−0.851013	−0.425506	−	0.904955i	$-0.639904\pi$
$937$	−24408.7	−0.851013	−0.425506	+	0.904955i	$0.639904\pi$
$938$	0	0
$939$	−29754.0	−1.03406
$940$	0	0
$941$	45350.3	1.57107	0.785536	−	0.618817i	$-0.212388\pi$
$941$	45350.3	1.57107	0.785536	+	0.618817i	$0.212388\pi$
$942$	0	0
$943$	−27453.5	−0.948049
$944$	0	0
$945$	−12848.7	−0.442295
$946$	0	0
$947$	−55992.0	−1.92133	−0.960663	−	0.277716i	$-0.910423\pi$
$947$	−55992.0	−1.92133	−0.960663	+	0.277716i	$0.910423\pi$
$948$	0	0
$949$	−20074.1	−0.686653
$950$	0	0
$951$	40607.4	1.38463
$952$	0	0
$953$	−21748.8	−0.739259	−0.369629	−	0.929179i	$-0.620515\pi$
$953$	−21748.8	−0.739259	−0.369629	+	0.929179i	$0.620515\pi$
$954$	0	0
$955$	−9926.28	−0.336342
$956$	0	0
$957$	1848.78	0.0624478
$958$	0	0
$959$	−55155.0	−1.85719
$960$	0	0
$961$	−8388.63	−0.281583
$962$	0	0
$963$	12814.4	0.428804
$964$	0	0
$965$	−14798.5	−0.493657
$966$	0	0
$967$	−16210.9	−0.539099	−0.269549	−	0.962987i	$-0.586875\pi$
$967$	−16210.9	−0.539099	−0.269549	+	0.962987i	$0.586875\pi$
$968$	0	0
$969$	−22809.9	−0.756202
$970$	0	0
$971$	−36637.1	−1.21085	−0.605427	−	0.795900i	$-0.706998\pi$
$971$	−36637.1	−1.21085	−0.605427	+	0.795900i	$0.706998\pi$
$972$	0	0
$973$	58142.1	1.91567
$974$	0	0
$975$	37452.4	1.23019
$976$	0	0
$977$	45374.0	1.48582	0.742908	−	0.669394i	$-0.233446\pi$
$977$	45374.0	1.48582	0.742908	+	0.669394i	$0.233446\pi$
$978$	0	0
$979$	15942.9	0.520467
$980$	0	0
$981$	9774.88	0.318132
$982$	0	0
$983$	3947.58	0.128086	0.0640428	−	0.997947i	$-0.479601\pi$
$983$	3947.58	0.128086	0.0640428	+	0.997947i	$0.479601\pi$
$984$	0	0
$985$	−7929.45	−0.256501
$986$	0	0
$987$	345.945	0.0111566
$988$	0	0
$989$	−41288.2	−1.32749
$990$	0	0
$991$	−60573.4	−1.94165	−0.970826	−	0.239784i	$-0.922923\pi$
$991$	−60573.4	−1.94165	−0.970826	+	0.239784i	$0.922923\pi$
$992$	0	0
$993$	23139.6	0.739489
$994$	0	0
$995$	15285.1	0.487006
$996$	0	0
$997$	11461.1	0.364069	0.182035	−	0.983292i	$-0.441732\pi$
$997$	11461.1	0.364069	0.182035	+	0.983292i	$0.441732\pi$
$998$	0	0
$999$	−11651.5	−0.369005

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1856.4.a.bk.1.9		12
4.3	odd	2	inner	1856.4.a.bk.1.4		12
8.3	odd	2		928.4.a.i.1.9	yes	12
8.5	even	2		928.4.a.i.1.4	✓	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
928.4.a.i.1.4	✓	12	8.5	even	2
928.4.a.i.1.9	yes	12	8.3	odd	2
1856.4.a.bk.1.4		12	4.3	odd	2	inner
1856.4.a.bk.1.9		12	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)

\(f(q)\)	\(=\)	\(q+3.82226 q^{3} -3.68395 q^{5} -23.1652 q^{7} -12.3903 q^{9} +O(q^{10})\)	\(q+3.82226 q^{3} -3.68395 q^{5} -23.1652 q^{7} -12.3903 q^{9} +16.6789 q^{11} -87.9352 q^{13} -14.0810 q^{15} -58.1595 q^{17} +102.608 q^{19} -88.5436 q^{21} -111.738 q^{23} -111.429 q^{25} -150.560 q^{27} +29.0000 q^{29} +146.296 q^{31} +63.7510 q^{33} +85.3395 q^{35} +77.3874 q^{37} -336.111 q^{39} +245.696 q^{41} +369.510 q^{43} +45.6454 q^{45} -3.90706 q^{47} +193.629 q^{49} -222.301 q^{51} -355.978 q^{53} -61.4441 q^{55} +392.196 q^{57} -223.920 q^{59} +308.765 q^{61} +287.025 q^{63} +323.948 q^{65} -548.837 q^{67} -427.091 q^{69} -121.875 q^{71} +228.283 q^{73} -425.909 q^{75} -386.371 q^{77} +665.320 q^{79} -240.940 q^{81} +1417.96 q^{83} +214.256 q^{85} +110.845 q^{87} +955.874 q^{89} +2037.04 q^{91} +559.179 q^{93} -378.004 q^{95} -598.996 q^{97} -206.657 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 10 q^{5} + 166 q^{9}+O(q^{10}) \) 12 * q - 10 * q^5 + 166 * q^9	\( 12 q - 10 q^{5} + 166 q^{9} - 30 q^{13} + 336 q^{17} - 56 q^{21} + 294 q^{25} + 348 q^{29} + 214 q^{33} - 196 q^{37} + 940 q^{41} - 1672 q^{45} + 972 q^{49} + 406 q^{53} + 1224 q^{57} - 400 q^{61} + 2998 q^{65} - 1004 q^{69} + 2252 q^{73} - 3032 q^{77} + 3932 q^{81} - 5564 q^{85} + 5212 q^{89} - 3722 q^{93} + 6164 q^{97}+O(q^{100}) \) 12 * q - 10 * q^5 + 166 * q^9 - 30 * q^13 + 336 * q^17 - 56 * q^21 + 294 * q^25 + 348 * q^29 + 214 * q^33 - 196 * q^37 + 940 * q^41 - 1672 * q^45 + 972 * q^49 + 406 * q^53 + 1224 * q^57 - 400 * q^61 + 2998 * q^65 - 1004 * q^69 + 2252 * q^73 - 3032 * q^77 + 3932 * q^81 - 5564 * q^85 + 5212 * q^89 - 3722 * q^93 + 6164 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more