LMFDB - Embedded newform 928.4.a.e.1.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [928,4,Mod(1,928)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(928, 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("928.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 928 = 2^{5} \cdot 29 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	928.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:	$54.7537724853$
Analytic rank:	$0$
Dimension:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - 4x^{8} - 138x^{7} + 394x^{6} + 5872x^{5} - 10822x^{4} - 85158x^{3} + 30654x^{2} + 439999x + 396802 $ x^9 - 4x^8 - 138x^7 + 394x^6 + 5872x^5 - 10822x^4 - 85158x^3 + 30654x^2 + 439999x + 396802
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{12} $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Root			$6.50540$ of defining polynomial
Character	$\chi$	$=$	928.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+6.50540 q^{3} -1.74434 q^{5} +26.0581 q^{7} +15.3202 q^{9} +O(q^{10})$	$q+6.50540 q^{3} -1.74434 q^{5} +26.0581 q^{7} +15.3202 q^{9} +34.5346 q^{11} +43.9865 q^{13} -11.3476 q^{15} +134.214 q^{17} -39.5628 q^{19} +169.519 q^{21} -22.5760 q^{23} -121.957 q^{25} -75.9817 q^{27} -29.0000 q^{29} -90.1691 q^{31} +224.661 q^{33} -45.4542 q^{35} +213.354 q^{37} +286.150 q^{39} -328.864 q^{41} +2.01802 q^{43} -26.7236 q^{45} +16.3670 q^{47} +336.027 q^{49} +873.113 q^{51} +149.430 q^{53} -60.2400 q^{55} -257.372 q^{57} -292.706 q^{59} -111.495 q^{61} +399.216 q^{63} -76.7273 q^{65} +159.934 q^{67} -146.866 q^{69} +129.341 q^{71} +742.501 q^{73} -793.381 q^{75} +899.907 q^{77} +1064.85 q^{79} -907.937 q^{81} -496.664 q^{83} -234.114 q^{85} -188.657 q^{87} +676.600 q^{89} +1146.21 q^{91} -586.586 q^{93} +69.0110 q^{95} -103.230 q^{97} +529.077 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q + 4 q^{3} + 10 q^{5} + 12 q^{7} + 49 q^{9}+O(q^{10}) $ 9 * q + 4 * q^3 + 10 * q^5 + 12 * q^7 + 49 * q^9	$ 9 q + 4 q^{3} + 10 q^{5} + 12 q^{7} + 49 q^{9} + 64 q^{11} + 70 q^{13} + 170 q^{15} - 66 q^{17} + 42 q^{19} + 76 q^{21} + 40 q^{23} + 111 q^{25} + 322 q^{27} - 261 q^{29} - 64 q^{31} - 52 q^{33} + 496 q^{35} - 54 q^{37} + 590 q^{39} - 378 q^{41} - 32 q^{43} + 1046 q^{45} + 1164 q^{47} - 351 q^{49} + 376 q^{51} + 278 q^{53} + 614 q^{55} + 28 q^{57} + 640 q^{59} + 1054 q^{61} + 1660 q^{63} - 708 q^{65} + 1184 q^{67} + 188 q^{69} + 1988 q^{71} - 750 q^{73} + 3126 q^{75} + 1260 q^{77} + 2916 q^{79} + 293 q^{81} + 2832 q^{83} + 56 q^{85} - 116 q^{87} - 370 q^{89} + 3016 q^{91} - 1696 q^{93} + 4412 q^{95} - 2234 q^{97} + 4118 q^{99}+O(q^{100}) $ 9 * q + 4 * q^3 + 10 * q^5 + 12 * q^7 + 49 * q^9 + 64 * q^11 + 70 * q^13 + 170 * q^15 - 66 * q^17 + 42 * q^19 + 76 * q^21 + 40 * q^23 + 111 * q^25 + 322 * q^27 - 261 * q^29 - 64 * q^31 - 52 * q^33 + 496 * q^35 - 54 * q^37 + 590 * q^39 - 378 * q^41 - 32 * q^43 + 1046 * q^45 + 1164 * q^47 - 351 * q^49 + 376 * q^51 + 278 * q^53 + 614 * q^55 + 28 * q^57 + 640 * q^59 + 1054 * q^61 + 1660 * q^63 - 708 * q^65 + 1184 * q^67 + 188 * q^69 + 1988 * q^71 - 750 * q^73 + 3126 * q^75 + 1260 * q^77 + 2916 * q^79 + 293 * q^81 + 2832 * q^83 + 56 * q^85 - 116 * q^87 - 370 * q^89 + 3016 * q^91 - 1696 * q^93 + 4412 * q^95 - 2234 * q^97 + 4118 * q^99

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
$3$	6.50540	1.25196	0.625982	−	0.779837i	$-0.284698\pi$
$3$	6.50540	1.25196	0.625982	+	0.779837i	$0.284698\pi$
$4$	0	0
$5$	−1.74434	−0.156018	−0.0780092	−	0.996953i	$-0.524856\pi$
$5$	−1.74434	−0.156018	−0.0780092	+	0.996953i	$0.524856\pi$
$6$	0	0
$7$	26.0581	1.40701	0.703504	−	0.710692i	$-0.251618\pi$
$7$	26.0581	1.40701	0.703504	+	0.710692i	$0.251618\pi$
$8$	0	0
$9$	15.3202	0.567415
$10$	0	0
$11$	34.5346	0.946597	0.473298	−	0.880902i	$-0.343063\pi$
$11$	34.5346	0.946597	0.473298	+	0.880902i	$0.343063\pi$
$12$	0	0
$13$	43.9865	0.938435	0.469218	−	0.883083i	$-0.344536\pi$
$13$	43.9865	0.938435	0.469218	+	0.883083i	$0.344536\pi$
$14$	0	0
$15$	−11.3476	−0.195329
$16$	0	0
$17$	134.214	1.91480	0.957400	−	0.288766i	$-0.0932448\pi$
$17$	134.214	1.91480	0.957400	+	0.288766i	$0.0932448\pi$
$18$	0	0
$19$	−39.5628	−0.477702	−0.238851	−	0.971056i	$-0.576771\pi$
$19$	−39.5628	−0.477702	−0.238851	+	0.971056i	$0.576771\pi$
$20$	0	0
$21$	169.519	1.76152
$22$	0	0
$23$	−22.5760	−0.204671	−0.102335	−	0.994750i	$-0.532632\pi$
$23$	−22.5760	−0.204671	−0.102335	+	0.994750i	$0.532632\pi$
$24$	0	0
$25$	−121.957	−0.975658
$26$	0	0
$27$	−75.9817	−0.541581
$28$	0	0
$29$	−29.0000	−0.185695
$29$	−29.0000	−0.185695
$30$	0	0
$31$	−90.1691	−0.522414	−0.261207	−	0.965283i	$-0.584121\pi$
$31$	−90.1691	−0.522414	−0.261207	+	0.965283i	$0.584121\pi$
$32$	0	0
$33$	224.661	1.18511
$34$	0	0
$35$	−45.4542	−0.219519
$36$	0	0
$37$	213.354	0.947976	0.473988	−	0.880531i	$-0.342814\pi$
$37$	213.354	0.947976	0.473988	+	0.880531i	$0.342814\pi$
$38$	0	0
$39$	286.150	1.17489
$40$	0	0
$41$	−328.864	−1.25268	−0.626340	−	0.779550i	$-0.715448\pi$
$41$	−328.864	−1.25268	−0.626340	+	0.779550i	$0.715448\pi$
$42$	0	0
$43$	2.01802	0.00715686	0.00357843	−	0.999994i	$-0.498861\pi$
$43$	2.01802	0.00715686	0.00357843	+	0.999994i	$0.498861\pi$
$44$	0	0
$45$	−26.7236	−0.0885272
$46$	0	0
$47$	16.3670	0.0507952	0.0253976	−	0.999677i	$-0.491915\pi$
$47$	16.3670	0.0507952	0.0253976	+	0.999677i	$0.491915\pi$
$48$	0	0
$49$	336.027	0.979670
$50$	0	0
$51$	873.113	2.39726
$52$	0	0
$53$	149.430	0.387280	0.193640	−	0.981073i	$-0.437971\pi$
$53$	149.430	0.387280	0.193640	+	0.981073i	$0.437971\pi$
$54$	0	0
$55$	−60.2400	−0.147686
$56$	0	0
$57$	−257.372	−0.598066
$58$	0	0
$59$	−292.706	−0.645882	−0.322941	−	0.946419i	$-0.604672\pi$
$59$	−292.706	−0.645882	−0.322941	+	0.946419i	$0.604672\pi$
$60$	0	0
$61$	−111.495	−0.234024	−0.117012	−	0.993131i	$-0.537332\pi$
$61$	−111.495	−0.234024	−0.117012	+	0.993131i	$0.537332\pi$
$62$	0	0
$63$	399.216	0.798358
$64$	0	0
$65$	−76.7273	−0.146413
$66$	0	0
$67$	159.934	0.291628	0.145814	−	0.989312i	$-0.453420\pi$
$67$	159.934	0.291628	0.145814	+	0.989312i	$0.453420\pi$
$68$	0	0
$69$	−146.866	−0.256241
$70$	0	0
$71$	129.341	0.216196	0.108098	−	0.994140i	$-0.465524\pi$
$71$	129.341	0.216196	0.108098	+	0.994140i	$0.465524\pi$
$72$	0	0
$73$	742.501	1.19045	0.595227	−	0.803558i	$-0.297062\pi$
$73$	742.501	1.19045	0.595227	+	0.803558i	$0.297062\pi$
$74$	0	0
$75$	−793.381	−1.22149
$76$	0	0
$77$	899.907	1.33187
$78$	0	0
$79$	1064.85	1.51651	0.758257	−	0.651956i	$-0.226051\pi$
$79$	1064.85	1.51651	0.758257	+	0.651956i	$0.226051\pi$
$80$	0	0
$81$	−907.937	−1.24546
$82$	0	0
$83$	−496.664	−0.656818	−0.328409	−	0.944536i	$-0.606513\pi$
$83$	−496.664	−0.656818	−0.328409	+	0.944536i	$0.606513\pi$
$84$	0	0
$85$	−234.114	−0.298744
$86$	0	0
$87$	−188.657	−0.232484
$88$	0	0
$89$	676.600	0.805836	0.402918	−	0.915236i	$-0.367996\pi$
$89$	676.600	0.805836	0.402918	+	0.915236i	$0.367996\pi$
$90$	0	0
$91$	1146.21	1.32039
$92$	0	0
$93$	−586.586	−0.654044
$94$	0	0
$95$	69.0110	0.0745303
$96$	0	0
$97$	−103.230	−0.108056	−0.0540278	−	0.998539i	$-0.517206\pi$
$97$	−103.230	−0.108056	−0.0540278	+	0.998539i	$0.517206\pi$
$98$	0	0
$99$	529.077	0.537113
$100$	0	0
$101$	1396.34	1.37566	0.687828	−	0.725874i	$-0.258564\pi$
$101$	1396.34	1.37566	0.687828	+	0.725874i	$0.258564\pi$
$102$	0	0
$103$	1828.75	1.74944	0.874719	−	0.484630i	$-0.161046\pi$
$103$	1828.75	1.74944	0.874719	+	0.484630i	$0.161046\pi$
$104$	0	0
$105$	−295.698	−0.274830
$106$	0	0
$107$	1116.86	1.00907	0.504536	−	0.863390i	$-0.331663\pi$
$107$	1116.86	1.00907	0.504536	+	0.863390i	$0.331663\pi$
$108$	0	0
$109$	−598.302	−0.525751	−0.262876	−	0.964830i	$-0.584671\pi$
$109$	−598.302	−0.525751	−0.262876	+	0.964830i	$0.584671\pi$
$110$	0	0
$111$	1387.95	1.18683
$112$	0	0
$113$	−2090.66	−1.74046	−0.870232	−	0.492643i	$-0.836031\pi$
$113$	−2090.66	−1.74046	−0.870232	+	0.492643i	$0.836031\pi$
$114$	0	0
$115$	39.3803	0.0319324
$116$	0	0
$117$	673.883	0.532483
$118$	0	0
$119$	3497.36	2.69414
$120$	0	0
$121$	−138.364	−0.103955
$122$	0	0
$123$	−2139.39	−1.56831
$124$	0	0
$125$	430.777	0.308239
$126$	0	0
$127$	−1767.83	−1.23519	−0.617597	−	0.786495i	$-0.711894\pi$
$127$	−1767.83	−1.23519	−0.617597	+	0.786495i	$0.711894\pi$
$128$	0	0
$129$	13.1280	0.00896013
$130$	0	0
$131$	−1823.75	−1.21635	−0.608174	−	0.793804i	$-0.708098\pi$
$131$	−1823.75	−1.21635	−0.608174	+	0.793804i	$0.708098\pi$
$132$	0	0
$133$	−1030.93	−0.672130
$134$	0	0
$135$	132.538	0.0844965
$136$	0	0
$137$	−2417.52	−1.50761	−0.753806	−	0.657097i	$-0.771784\pi$
$137$	−2417.52	−1.50761	−0.753806	+	0.657097i	$0.771784\pi$
$138$	0	0
$139$	−1787.82	−1.09094	−0.545470	−	0.838130i	$-0.683649\pi$
$139$	−1787.82	−1.09094	−0.545470	+	0.838130i	$0.683649\pi$
$140$	0	0
$141$	106.474	0.0635938
$142$	0	0
$143$	1519.05	0.888320
$144$	0	0
$145$	50.5858	0.0289719
$146$	0	0
$147$	2185.99	1.22651
$148$	0	0
$149$	−348.880	−0.191821	−0.0959106	−	0.995390i	$-0.530576\pi$
$149$	−348.880	−0.191821	−0.0959106	+	0.995390i	$0.530576\pi$
$150$	0	0
$151$	1024.52	0.552147	0.276073	−	0.961137i	$-0.410967\pi$
$151$	1024.52	0.552147	0.276073	+	0.961137i	$0.410967\pi$
$152$	0	0
$153$	2056.18	1.08649
$154$	0	0
$155$	157.285	0.0815062
$156$	0	0
$157$	−920.398	−0.467871	−0.233935	−	0.972252i	$-0.575160\pi$
$157$	−920.398	−0.467871	−0.233935	+	0.972252i	$0.575160\pi$
$158$	0	0
$159$	972.104	0.484861
$160$	0	0
$161$	−588.290	−0.287974
$162$	0	0
$163$	1214.70	0.583696	0.291848	−	0.956465i	$-0.405730\pi$
$163$	1214.70	0.583696	0.291848	+	0.956465i	$0.405730\pi$
$164$	0	0
$165$	−391.885	−0.184898
$166$	0	0
$167$	3724.63	1.72587	0.862935	−	0.505315i	$-0.168624\pi$
$167$	3724.63	1.72587	0.862935	+	0.505315i	$0.168624\pi$
$168$	0	0
$169$	−262.188	−0.119339
$170$	0	0
$171$	−606.111	−0.271056
$172$	0	0
$173$	1251.03	0.549792	0.274896	−	0.961474i	$-0.411357\pi$
$173$	1251.03	0.549792	0.274896	+	0.961474i	$0.411357\pi$
$174$	0	0
$175$	−3177.98	−1.37276
$176$	0	0
$177$	−1904.17	−0.808622
$178$	0	0
$179$	3162.67	1.32061	0.660305	−	0.750998i	$-0.270427\pi$
$179$	3162.67	1.32061	0.660305	+	0.750998i	$0.270427\pi$
$180$	0	0
$181$	−2922.50	−1.20015	−0.600076	−	0.799943i	$-0.704863\pi$
$181$	−2922.50	−1.20015	−0.600076	+	0.799943i	$0.704863\pi$
$182$	0	0
$183$	−725.319	−0.292990
$184$	0	0
$185$	−372.161	−0.147902
$186$	0	0
$187$	4635.01	1.81254
$188$	0	0
$189$	−1979.94	−0.762008
$190$	0	0
$191$	−220.312	−0.0834620	−0.0417310	−	0.999129i	$-0.513287\pi$
$191$	−220.312	−0.0834620	−0.0417310	+	0.999129i	$0.513287\pi$
$192$	0	0
$193$	−3256.89	−1.21470	−0.607348	−	0.794436i	$-0.707767\pi$
$193$	−3256.89	−1.21470	−0.607348	+	0.794436i	$0.707767\pi$
$194$	0	0
$195$	−499.142	−0.183304
$196$	0	0
$197$	3771.73	1.36409	0.682043	−	0.731312i	$-0.261092\pi$
$197$	3771.73	1.36409	0.682043	+	0.731312i	$0.261092\pi$
$198$	0	0
$199$	−2497.30	−0.889593	−0.444797	−	0.895632i	$-0.646724\pi$
$199$	−2497.30	−0.889593	−0.444797	+	0.895632i	$0.646724\pi$
$200$	0	0
$201$	1040.44	0.365108
$202$	0	0
$203$	−755.686	−0.261275
$204$	0	0
$205$	573.650	0.195441
$206$	0	0
$207$	−345.870	−0.116133
$208$	0	0
$209$	−1366.29	−0.452191
$210$	0	0
$211$	−913.180	−0.297943	−0.148971	−	0.988842i	$-0.547596\pi$
$211$	−913.180	−0.297943	−0.148971	+	0.988842i	$0.547596\pi$
$212$	0	0
$213$	841.415	0.270670
$214$	0	0
$215$	−3.52011	−0.00111660
$216$	0	0
$217$	−2349.64	−0.735041
$218$	0	0
$219$	4830.26	1.49041
$220$	0	0
$221$	5903.59	1.79692
$222$	0	0
$223$	−4122.81	−1.23805	−0.619023	−	0.785373i	$-0.712471\pi$
$223$	−4122.81	−1.23805	−0.619023	+	0.785373i	$0.712471\pi$
$224$	0	0
$225$	−1868.41	−0.553604
$226$	0	0
$227$	3632.84	1.06220	0.531102	−	0.847308i	$-0.321778\pi$
$227$	3632.84	1.06220	0.531102	+	0.847308i	$0.321778\pi$
$228$	0	0
$229$	6428.20	1.85497	0.927484	−	0.373864i	$-0.121967\pi$
$229$	6428.20	1.85497	0.927484	+	0.373864i	$0.121967\pi$
$230$	0	0
$231$	5854.25	1.66745
$232$	0	0
$233$	−3844.90	−1.08106	−0.540532	−	0.841324i	$-0.681777\pi$
$233$	−3844.90	−1.08106	−0.540532	+	0.841324i	$0.681777\pi$
$234$	0	0
$235$	−28.5496	−0.00792498
$236$	0	0
$237$	6927.25	1.89862
$238$	0	0
$239$	−2434.71	−0.658947	−0.329473	−	0.944165i	$-0.606871\pi$
$239$	−2434.71	−0.658947	−0.329473	+	0.944165i	$0.606871\pi$
$240$	0	0
$241$	−4450.57	−1.18957	−0.594785	−	0.803885i	$-0.702763\pi$
$241$	−4450.57	−1.18957	−0.594785	+	0.803885i	$0.702763\pi$
$242$	0	0
$243$	−3854.99	−1.01769
$244$	0	0
$245$	−586.144	−0.152846
$246$	0	0
$247$	−1740.23	−0.448293
$248$	0	0
$249$	−3231.00	−0.822313
$250$	0	0
$251$	3603.45	0.906166	0.453083	−	0.891468i	$-0.350324\pi$
$251$	3603.45	0.906166	0.453083	+	0.891468i	$0.350324\pi$
$252$	0	0
$253$	−779.654	−0.193741
$254$	0	0
$255$	−1523.01	−0.374017
$256$	0	0
$257$	3169.32	0.769248	0.384624	−	0.923073i	$-0.374331\pi$
$257$	3169.32	0.769248	0.384624	+	0.923073i	$0.374331\pi$
$258$	0	0
$259$	5559.60	1.33381
$260$	0	0
$261$	−444.286	−0.105366
$262$	0	0
$263$	−901.230	−0.211301	−0.105651	−	0.994403i	$-0.533693\pi$
$263$	−901.230	−0.211301	−0.105651	+	0.994403i	$0.533693\pi$
$264$	0	0
$265$	−260.657	−0.0604228
$266$	0	0
$267$	4401.55	1.00888
$268$	0	0
$269$	3218.85	0.729578	0.364789	−	0.931090i	$-0.381141\pi$
$269$	3218.85	0.729578	0.364789	+	0.931090i	$0.381141\pi$
$270$	0	0
$271$	−8364.23	−1.87487	−0.937437	−	0.348154i	$-0.886809\pi$
$271$	−8364.23	−1.87487	−0.937437	+	0.348154i	$0.886809\pi$
$272$	0	0
$273$	7456.53	1.65308
$274$	0	0
$275$	−4211.74	−0.923555
$276$	0	0
$277$	2507.41	0.543882	0.271941	−	0.962314i	$-0.412334\pi$
$277$	2507.41	0.543882	0.271941	+	0.962314i	$0.412334\pi$
$278$	0	0
$279$	−1381.41	−0.296426
$280$	0	0
$281$	7813.51	1.65877	0.829386	−	0.558676i	$-0.188690\pi$
$281$	7813.51	1.65877	0.829386	+	0.558676i	$0.188690\pi$
$282$	0	0
$283$	7382.99	1.55079	0.775394	−	0.631478i	$-0.217551\pi$
$283$	7382.99	1.55079	0.775394	+	0.631478i	$0.217551\pi$
$284$	0	0
$285$	448.944	0.0933093
$286$	0	0
$287$	−8569.58	−1.76253
$288$	0	0
$289$	13100.3	2.66646
$290$	0	0
$291$	−671.551	−0.135282
$292$	0	0
$293$	−4876.95	−0.972405	−0.486203	−	0.873846i	$-0.661618\pi$
$293$	−4876.95	−0.972405	−0.486203	+	0.873846i	$0.661618\pi$
$294$	0	0
$295$	510.578	0.100770
$296$	0	0
$297$	−2623.99	−0.512658
$298$	0	0
$299$	−993.041	−0.192070
$300$	0	0
$301$	52.5858	0.0100698
$302$	0	0
$303$	9083.76	1.72227
$304$	0	0
$305$	194.485	0.0365120
$306$	0	0
$307$	−6722.28	−1.24971	−0.624854	−	0.780741i	$-0.714842\pi$
$307$	−6722.28	−1.24971	−0.624854	+	0.780741i	$0.714842\pi$
$308$	0	0
$309$	11896.8	2.19024
$310$	0	0
$311$	1684.02	0.307049	0.153524	−	0.988145i	$-0.450938\pi$
$311$	1684.02	0.307049	0.153524	+	0.988145i	$0.450938\pi$
$312$	0	0
$313$	−9547.82	−1.72420	−0.862100	−	0.506738i	$-0.830851\pi$
$313$	−9547.82	−1.72420	−0.862100	+	0.506738i	$0.830851\pi$
$314$	0	0
$315$	−696.368	−0.124558
$316$	0	0
$317$	−4080.15	−0.722915	−0.361457	−	0.932389i	$-0.617721\pi$
$317$	−4080.15	−0.722915	−0.361457	+	0.932389i	$0.617721\pi$
$318$	0	0
$319$	−1001.50	−0.175779
$320$	0	0
$321$	7265.61	1.26332
$322$	0	0
$323$	−5309.87	−0.914704
$324$	0	0
$325$	−5364.47	−0.915592
$326$	0	0
$327$	−3892.19	−0.658222
$328$	0	0
$329$	426.494	0.0714692
$330$	0	0
$331$	2013.41	0.334341	0.167171	−	0.985928i	$-0.446537\pi$
$331$	2013.41	0.334341	0.167171	+	0.985928i	$0.446537\pi$
$332$	0	0
$333$	3268.62	0.537896
$334$	0	0
$335$	−278.980	−0.0454993
$336$	0	0
$337$	−8373.60	−1.35353	−0.676764	−	0.736200i	$-0.736618\pi$
$337$	−8373.60	−1.35353	−0.676764	+	0.736200i	$0.736618\pi$
$338$	0	0
$339$	−13600.5	−2.17900
$340$	0	0
$341$	−3113.95	−0.494516
$342$	0	0
$343$	−181.708	−0.0286044
$344$	0	0
$345$	256.184	0.0399783
$346$	0	0
$347$	−6584.86	−1.01871	−0.509357	−	0.860555i	$-0.670117\pi$
$347$	−6584.86	−1.01871	−0.509357	+	0.860555i	$0.670117\pi$
$348$	0	0
$349$	−2267.59	−0.347798	−0.173899	−	0.984764i	$-0.555637\pi$
$349$	−2267.59	−0.347798	−0.173899	+	0.984764i	$0.555637\pi$
$350$	0	0
$351$	−3342.17	−0.508238
$352$	0	0
$353$	1025.12	0.154565	0.0772826	−	0.997009i	$-0.475376\pi$
$353$	1025.12	0.154565	0.0772826	+	0.997009i	$0.475376\pi$
$354$	0	0
$355$	−225.614	−0.0337306
$356$	0	0
$357$	22751.7	3.37296
$358$	0	0
$359$	1688.16	0.248183	0.124091	−	0.992271i	$-0.460398\pi$
$359$	1688.16	0.248183	0.124091	+	0.992271i	$0.460398\pi$
$360$	0	0
$361$	−5293.78	−0.771801
$362$	0	0
$363$	−900.112	−0.130148
$364$	0	0
$365$	−1295.17	−0.185733
$366$	0	0
$367$	192.611	0.0273957	0.0136978	−	0.999906i	$-0.495640\pi$
$367$	192.611	0.0273957	0.0136978	+	0.999906i	$0.495640\pi$
$368$	0	0
$369$	−5038.27	−0.710790
$370$	0	0
$371$	3893.88	0.544906
$372$	0	0
$373$	−6947.18	−0.964374	−0.482187	−	0.876068i	$-0.660157\pi$
$373$	−6947.18	−0.964374	−0.482187	+	0.876068i	$0.660157\pi$
$374$	0	0
$375$	2802.38	0.385904
$376$	0	0
$377$	−1275.61	−0.174263
$378$	0	0
$379$	7500.31	1.01653	0.508265	−	0.861200i	$-0.330287\pi$
$379$	7500.31	1.01653	0.508265	+	0.861200i	$0.330287\pi$
$380$	0	0
$381$	−11500.4	−1.54642
$382$	0	0
$383$	−7259.99	−0.968586	−0.484293	−	0.874906i	$-0.660923\pi$
$383$	−7259.99	−0.968586	−0.484293	+	0.874906i	$0.660923\pi$
$384$	0	0
$385$	−1569.74	−0.207796
$386$	0	0
$387$	30.9165	0.00406091
$388$	0	0
$389$	−3912.76	−0.509987	−0.254994	−	0.966943i	$-0.582073\pi$
$389$	−3912.76	−0.509987	−0.254994	+	0.966943i	$0.582073\pi$
$390$	0	0
$391$	−3030.01	−0.391904
$392$	0	0
$393$	−11864.2	−1.52283
$394$	0	0
$395$	−1857.45	−0.236604
$396$	0	0
$397$	3532.50	0.446577	0.223288	−	0.974752i	$-0.428321\pi$
$397$	3532.50	0.446577	0.223288	+	0.974752i	$0.428321\pi$
$398$	0	0
$399$	−6706.64	−0.841483
$400$	0	0
$401$	−9853.53	−1.22709	−0.613543	−	0.789661i	$-0.710256\pi$
$401$	−9853.53	−1.22709	−0.613543	+	0.789661i	$0.710256\pi$
$402$	0	0
$403$	−3966.22	−0.490252
$404$	0	0
$405$	1583.75	0.194314
$406$	0	0
$407$	7368.07	0.897351
$408$	0	0
$409$	−8972.45	−1.08474	−0.542370	−	0.840139i	$-0.682473\pi$
$409$	−8972.45	−1.08474	−0.542370	+	0.840139i	$0.682473\pi$
$410$	0	0
$411$	−15726.9	−1.88748
$412$	0	0
$413$	−7627.37	−0.908761
$414$	0	0
$415$	866.350	0.102476
$416$	0	0
$417$	−11630.5	−1.36582
$418$	0	0
$419$	−12190.9	−1.42139	−0.710697	−	0.703498i	$-0.751620\pi$
$419$	−12190.9	−1.42139	−0.710697	+	0.703498i	$0.751620\pi$
$420$	0	0
$421$	4044.75	0.468241	0.234120	−	0.972208i	$-0.424779\pi$
$421$	4044.75	0.468241	0.234120	+	0.972208i	$0.424779\pi$
$422$	0	0
$423$	250.746	0.0288220
$424$	0	0
$425$	−16368.3	−1.86819
$426$	0	0
$427$	−2905.35	−0.329273
$428$	0	0
$429$	9882.06	1.11214
$430$	0	0
$431$	14523.1	1.62309	0.811545	−	0.584290i	$-0.198627\pi$
$431$	14523.1	1.62309	0.811545	+	0.584290i	$0.198627\pi$
$432$	0	0
$433$	5205.13	0.577697	0.288848	−	0.957375i	$-0.406728\pi$
$433$	5205.13	0.577697	0.288848	+	0.957375i	$0.406728\pi$
$434$	0	0
$435$	329.081	0.0362718
$436$	0	0
$437$	893.173	0.0977717
$438$	0	0
$439$	−12554.6	−1.36492	−0.682458	−	0.730925i	$-0.739089\pi$
$439$	−12554.6	−1.36492	−0.682458	+	0.730925i	$0.739089\pi$
$440$	0	0
$441$	5148.00	0.555880
$442$	0	0
$443$	15143.8	1.62416	0.812082	−	0.583544i	$-0.198334\pi$
$443$	15143.8	1.62416	0.812082	+	0.583544i	$0.198334\pi$
$444$	0	0
$445$	−1180.22	−0.125725
$446$	0	0
$447$	−2269.60	−0.240153
$448$	0	0
$449$	2918.35	0.306738	0.153369	−	0.988169i	$-0.450988\pi$
$449$	2918.35	0.306738	0.153369	+	0.988169i	$0.450988\pi$
$450$	0	0
$451$	−11357.2	−1.18578
$452$	0	0
$453$	6664.91	0.691268
$454$	0	0
$455$	−1999.37	−0.206004
$456$	0	0
$457$	−9572.94	−0.979876	−0.489938	−	0.871757i	$-0.662981\pi$
$457$	−9572.94	−0.979876	−0.489938	+	0.871757i	$0.662981\pi$
$458$	0	0
$459$	−10197.8	−1.03702
$460$	0	0
$461$	−15933.7	−1.60977	−0.804885	−	0.593430i	$-0.797773\pi$
$461$	−15933.7	−1.60977	−0.804885	+	0.593430i	$0.797773\pi$
$462$	0	0
$463$	−104.965	−0.0105360	−0.00526798	−	0.999986i	$-0.501677\pi$
$463$	−104.965	−0.0105360	−0.00526798	+	0.999986i	$0.501677\pi$
$464$	0	0
$465$	1023.20	0.102043
$466$	0	0
$467$	138.716	0.0137452	0.00687259	−	0.999976i	$-0.497812\pi$
$467$	138.716	0.0137452	0.00687259	+	0.999976i	$0.497812\pi$
$468$	0	0
$469$	4167.59	0.410323
$470$	0	0
$471$	−5987.55	−0.585758
$472$	0	0
$473$	69.6914	0.00677466
$474$	0	0
$475$	4824.98	0.466074
$476$	0	0
$477$	2289.31	0.219749
$478$	0	0
$479$	−9798.67	−0.934682	−0.467341	−	0.884077i	$-0.654788\pi$
$479$	−9798.67	−0.934682	−0.467341	+	0.884077i	$0.654788\pi$
$480$	0	0
$481$	9384.68	0.889614
$482$	0	0
$483$	−3827.06	−0.360533
$484$	0	0
$485$	180.068	0.0168587
$486$	0	0
$487$	−8385.14	−0.780220	−0.390110	−	0.920768i	$-0.627563\pi$
$487$	−8385.14	−0.780220	−0.390110	+	0.920768i	$0.627563\pi$
$488$	0	0
$489$	7902.09	0.730766
$490$	0	0
$491$	3106.61	0.285539	0.142769	−	0.989756i	$-0.454399\pi$
$491$	3106.61	0.285539	0.142769	+	0.989756i	$0.454399\pi$
$492$	0	0
$493$	−3892.20	−0.355569
$494$	0	0
$495$	−922.889	−0.0837996
$496$	0	0
$497$	3370.39	0.304190
$498$	0	0
$499$	54.8571	0.00492132	0.00246066	−	0.999997i	$-0.499217\pi$
$499$	54.8571	0.00492132	0.00246066	+	0.999997i	$0.499217\pi$
$500$	0	0
$501$	24230.2	2.16073
$502$	0	0
$503$	16425.8	1.45604	0.728020	−	0.685556i	$-0.240441\pi$
$503$	16425.8	1.45604	0.728020	+	0.685556i	$0.240441\pi$
$504$	0	0
$505$	−2435.69	−0.214627
$506$	0	0
$507$	−1705.64	−0.149408
$508$	0	0
$509$	564.244	0.0491349	0.0245675	−	0.999698i	$-0.492179\pi$
$509$	564.244	0.0491349	0.0245675	+	0.999698i	$0.492179\pi$
$510$	0	0
$511$	19348.2	1.67498
$512$	0	0
$513$	3006.05	0.258714
$514$	0	0
$515$	−3189.96	−0.272945
$516$	0	0
$517$	565.228	0.0480826
$518$	0	0
$519$	8138.45	0.688320
$520$	0	0
$521$	11436.1	0.961657	0.480828	−	0.876815i	$-0.340336\pi$
$521$	11436.1	0.961657	0.480828	+	0.876815i	$0.340336\pi$
$522$	0	0
$523$	−9988.34	−0.835104	−0.417552	−	0.908653i	$-0.637112\pi$
$523$	−9988.34	−0.835104	−0.417552	+	0.908653i	$0.637112\pi$
$524$	0	0
$525$	−20674.0	−1.71865
$526$	0	0
$527$	−12101.9	−1.00032
$528$	0	0
$529$	−11657.3	−0.958110
$530$	0	0
$531$	−4484.32	−0.366484
$532$	0	0
$533$	−14465.6	−1.17556
$534$	0	0
$535$	−1948.18	−0.157434
$536$	0	0
$537$	20574.4	1.65336
$538$	0	0
$539$	11604.5	0.927352
$540$	0	0
$541$	1900.85	0.151061	0.0755303	−	0.997144i	$-0.475935\pi$
$541$	1900.85	0.151061	0.0755303	+	0.997144i	$0.475935\pi$
$542$	0	0
$543$	−19012.0	−1.50255
$544$	0	0
$545$	1043.64	0.0820269
$546$	0	0
$547$	−9310.74	−0.727785	−0.363892	−	0.931441i	$-0.618552\pi$
$547$	−9310.74	−0.727785	−0.363892	+	0.931441i	$0.618552\pi$
$548$	0	0
$549$	−1708.13	−0.132789
$550$	0	0
$551$	1147.32	0.0887070
$552$	0	0
$553$	27747.9	2.13375
$554$	0	0
$555$	−2421.05	−0.185168
$556$	0	0
$557$	−10861.6	−0.826250	−0.413125	−	0.910674i	$-0.635563\pi$
$557$	−10861.6	−0.826250	−0.413125	+	0.910674i	$0.635563\pi$
$558$	0	0
$559$	88.7655	0.00671625
$560$	0	0
$561$	30152.6	2.26924
$562$	0	0
$563$	246.144	0.0184258	0.00921290	−	0.999958i	$-0.497067\pi$
$563$	246.144	0.0184258	0.00921290	+	0.999958i	$0.497067\pi$
$564$	0	0
$565$	3646.81	0.271544
$566$	0	0
$567$	−23659.1	−1.75236
$568$	0	0
$569$	1180.19	0.0869527	0.0434763	−	0.999054i	$-0.486157\pi$
$569$	1180.19	0.0869527	0.0434763	+	0.999054i	$0.486157\pi$
$570$	0	0
$571$	6841.42	0.501409	0.250704	−	0.968064i	$-0.419338\pi$
$571$	6841.42	0.501409	0.250704	+	0.968064i	$0.419338\pi$
$572$	0	0
$573$	−1433.22	−0.104492
$574$	0	0
$575$	2753.31	0.199689
$576$	0	0
$577$	−6422.86	−0.463409	−0.231705	−	0.972786i	$-0.574430\pi$
$577$	−6422.86	−0.463409	−0.231705	+	0.972786i	$0.574430\pi$
$578$	0	0
$579$	−21187.4	−1.52076
$580$	0	0
$581$	−12942.1	−0.924149
$582$	0	0
$583$	5160.51	0.366598
$584$	0	0
$585$	−1175.48	−0.0830771
$586$	0	0
$587$	16445.0	1.15632	0.578158	−	0.815925i	$-0.303772\pi$
$587$	16445.0	1.15632	0.578158	+	0.815925i	$0.303772\pi$
$588$	0	0
$589$	3567.34	0.249558
$590$	0	0
$591$	24536.6	1.70779
$592$	0	0
$593$	−13958.9	−0.966652	−0.483326	−	0.875440i	$-0.660571\pi$
$593$	−13958.9	−0.966652	−0.483326	+	0.875440i	$0.660571\pi$
$594$	0	0
$595$	−6100.58	−0.420335
$596$	0	0
$597$	−16245.9	−1.11374
$598$	0	0
$599$	−1007.54	−0.0687263	−0.0343632	−	0.999409i	$-0.510940\pi$
$599$	−1007.54	−0.0687263	−0.0343632	+	0.999409i	$0.510940\pi$
$600$	0	0
$601$	−3198.61	−0.217095	−0.108548	−	0.994091i	$-0.534620\pi$
$601$	−3198.61	−0.217095	−0.108548	+	0.994091i	$0.534620\pi$
$602$	0	0
$603$	2450.23	0.165474
$604$	0	0
$605$	241.353	0.0162189
$606$	0	0
$607$	−17511.7	−1.17097	−0.585483	−	0.810684i	$-0.699095\pi$
$607$	−17511.7	−1.17097	−0.585483	+	0.810684i	$0.699095\pi$
$608$	0	0
$609$	−4916.04	−0.327107
$610$	0	0
$611$	719.928	0.0476680
$612$	0	0
$613$	−2559.74	−0.168657	−0.0843285	−	0.996438i	$-0.526875\pi$
$613$	−2559.74	−0.168657	−0.0843285	+	0.996438i	$0.526875\pi$
$614$	0	0
$615$	3731.82	0.244685
$616$	0	0
$617$	−9033.40	−0.589418	−0.294709	−	0.955587i	$-0.595223\pi$
$617$	−9033.40	−0.589418	−0.294709	+	0.955587i	$0.595223\pi$
$618$	0	0
$619$	17108.3	1.11089	0.555445	−	0.831553i	$-0.312548\pi$
$619$	17108.3	1.11089	0.555445	+	0.831553i	$0.312548\pi$
$620$	0	0
$621$	1715.37	0.110846
$622$	0	0
$623$	17630.9	1.13382
$624$	0	0
$625$	14493.2	0.927567
$626$	0	0
$627$	−8888.23	−0.566127
$628$	0	0
$629$	28635.0	1.81518
$630$	0	0
$631$	−10191.7	−0.642987	−0.321494	−	0.946912i	$-0.604185\pi$
$631$	−10191.7	−0.642987	−0.321494	+	0.946912i	$0.604185\pi$
$632$	0	0
$633$	−5940.60	−0.373014
$634$	0	0
$635$	3083.69	0.192713
$636$	0	0
$637$	14780.6	0.919357
$638$	0	0
$639$	1981.53	0.122673
$640$	0	0
$641$	−15069.3	−0.928551	−0.464275	−	0.885691i	$-0.653685\pi$
$641$	−15069.3	−0.928551	−0.464275	+	0.885691i	$0.653685\pi$
$642$	0	0
$643$	25251.4	1.54871	0.774354	−	0.632752i	$-0.218075\pi$
$643$	25251.4	1.54871	0.774354	+	0.632752i	$0.218075\pi$
$644$	0	0
$645$	−22.8997	−0.00139794
$646$	0	0
$647$	13049.6	0.792941	0.396470	−	0.918047i	$-0.370235\pi$
$647$	13049.6	0.792941	0.396470	+	0.918047i	$0.370235\pi$
$648$	0	0
$649$	−10108.5	−0.611390
$650$	0	0
$651$	−15285.3	−0.920245
$652$	0	0
$653$	1741.94	0.104391	0.0521955	−	0.998637i	$-0.483378\pi$
$653$	1741.94	0.104391	0.0521955	+	0.998637i	$0.483378\pi$
$654$	0	0
$655$	3181.23	0.189773
$656$	0	0
$657$	11375.3	0.675482
$658$	0	0
$659$	29549.0	1.74669	0.873343	−	0.487105i	$-0.161947\pi$
$659$	29549.0	1.74669	0.873343	+	0.487105i	$0.161947\pi$
$660$	0	0
$661$	18000.8	1.05923	0.529615	−	0.848238i	$-0.322337\pi$
$661$	18000.8	1.05923	0.529615	+	0.848238i	$0.322337\pi$
$662$	0	0
$663$	38405.2	2.24967
$664$	0	0
$665$	1798.30	0.104865
$666$	0	0
$667$	654.705	0.0380064
$668$	0	0
$669$	−26820.6	−1.54999
$670$	0	0
$671$	−3850.43	−0.221526
$672$	0	0
$673$	6142.86	0.351842	0.175921	−	0.984404i	$-0.443710\pi$
$673$	6142.86	0.351842	0.175921	+	0.984404i	$0.443710\pi$
$674$	0	0
$675$	9266.52	0.528398
$676$	0	0
$677$	−26970.2	−1.53109	−0.765546	−	0.643381i	$-0.777531\pi$
$677$	−26970.2	−1.53109	−0.765546	+	0.643381i	$0.777531\pi$
$678$	0	0
$679$	−2689.98	−0.152035
$680$	0	0
$681$	23633.1	1.32984
$682$	0	0
$683$	−11702.2	−0.655598	−0.327799	−	0.944748i	$-0.606307\pi$
$683$	−11702.2	−0.655598	−0.327799	+	0.944748i	$0.606307\pi$
$684$	0	0
$685$	4216.98	0.235215
$686$	0	0
$687$	41818.0	2.32235
$688$	0	0
$689$	6572.92	0.363437
$690$	0	0
$691$	−22034.0	−1.21305	−0.606523	−	0.795066i	$-0.707436\pi$
$691$	−22034.0	−1.21305	−0.606523	+	0.795066i	$0.707436\pi$
$692$	0	0
$693$	13786.8	0.755723
$694$	0	0
$695$	3118.56	0.170207
$696$	0	0
$697$	−44138.0	−2.39863
$698$	0	0
$699$	−25012.6	−1.35345
$700$	0	0
$701$	26465.4	1.42594	0.712969	−	0.701195i	$-0.247350\pi$
$701$	26465.4	1.42594	0.712969	+	0.701195i	$0.247350\pi$
$702$	0	0
$703$	−8440.87	−0.452850
$704$	0	0
$705$	−185.727	−0.00992180
$706$	0	0
$707$	36386.1	1.93556
$708$	0	0
$709$	−3909.36	−0.207079	−0.103540	−	0.994625i	$-0.533017\pi$
$709$	−3909.36	−0.207079	−0.103540	+	0.994625i	$0.533017\pi$
$710$	0	0
$711$	16313.7	0.860493
$712$	0	0
$713$	2035.66	0.106923
$714$	0	0
$715$	−2649.74	−0.138594
$716$	0	0
$717$	−15838.8	−0.824978
$718$	0	0
$719$	27137.8	1.40761	0.703803	−	0.710395i	$-0.251484\pi$
$719$	27137.8	1.40761	0.703803	+	0.710395i	$0.251484\pi$
$720$	0	0
$721$	47653.9	2.46147
$722$	0	0
$723$	−28952.7	−1.48930
$724$	0	0
$725$	3536.76	0.181175
$726$	0	0
$727$	−23333.1	−1.19034	−0.595169	−	0.803600i	$-0.702915\pi$
$727$	−23333.1	−1.19034	−0.595169	+	0.803600i	$0.702915\pi$
$728$	0	0
$729$	−563.931	−0.0286507
$730$	0	0
$731$	270.846	0.0137039
$732$	0	0
$733$	19651.7	0.990248	0.495124	−	0.868822i	$-0.335123\pi$
$733$	19651.7	0.990248	0.495124	+	0.868822i	$0.335123\pi$
$734$	0	0
$735$	−3813.10	−0.191358
$736$	0	0
$737$	5523.26	0.276054
$738$	0	0
$739$	2012.48	0.100176	0.0500882	−	0.998745i	$-0.484050\pi$
$739$	2012.48	0.100176	0.0500882	+	0.998745i	$0.484050\pi$
$740$	0	0
$741$	−11320.9	−0.561246
$742$	0	0
$743$	−14171.2	−0.699720	−0.349860	−	0.936802i	$-0.613771\pi$
$743$	−14171.2	−0.699720	−0.349860	+	0.936802i	$0.613771\pi$
$744$	0	0
$745$	608.564	0.0299276
$746$	0	0
$747$	−7609.00	−0.372689
$748$	0	0
$749$	29103.3	1.41977
$750$	0	0
$751$	−8500.52	−0.413034	−0.206517	−	0.978443i	$-0.566213\pi$
$751$	−8500.52	−0.413034	−0.206517	+	0.978443i	$0.566213\pi$
$752$	0	0
$753$	23441.9	1.13449
$754$	0	0
$755$	−1787.11	−0.0861450
$756$	0	0
$757$	−15227.4	−0.731109	−0.365554	−	0.930790i	$-0.619121\pi$
$757$	−15227.4	−0.731109	−0.365554	+	0.930790i	$0.619121\pi$
$758$	0	0
$759$	−5071.96	−0.242557
$760$	0	0
$761$	−9243.82	−0.440326	−0.220163	−	0.975463i	$-0.570659\pi$
$761$	−9243.82	−0.440326	−0.220163	+	0.975463i	$0.570659\pi$
$762$	0	0
$763$	−15590.6	−0.739736
$764$	0	0
$765$	−3586.68	−0.169512
$766$	0	0
$767$	−12875.1	−0.606119
$768$	0	0
$769$	418.701	0.0196343	0.00981714	−	0.999952i	$-0.496875\pi$
$769$	418.701	0.0196343	0.00981714	+	0.999952i	$0.496875\pi$
$770$	0	0
$771$	20617.7	0.963071
$772$	0	0
$773$	−27488.2	−1.27902	−0.639508	−	0.768784i	$-0.720862\pi$
$773$	−27488.2	−1.27902	−0.639508	+	0.768784i	$0.720862\pi$
$774$	0	0
$775$	10996.8	0.509698
$776$	0	0
$777$	36167.4	1.66988
$778$	0	0
$779$	13010.8	0.598408
$780$	0	0
$781$	4466.73	0.204651
$782$	0	0
$783$	2203.47	0.100569
$784$	0	0
$785$	1605.48	0.0729965
$786$	0	0
$787$	7061.33	0.319834	0.159917	−	0.987130i	$-0.448877\pi$
$787$	7061.33	0.319834	0.159917	+	0.987130i	$0.448877\pi$
$788$	0	0
$789$	−5862.86	−0.264542
$790$	0	0
$791$	−54478.6	−2.44884
$792$	0	0
$793$	−4904.27	−0.219616
$794$	0	0
$795$	−1695.68	−0.0756472
$796$	0	0
$797$	−33174.1	−1.47439	−0.737195	−	0.675680i	$-0.763850\pi$
$797$	−33174.1	−1.47439	−0.737195	+	0.675680i	$0.763850\pi$
$798$	0	0
$799$	2196.68	0.0972626
$800$	0	0
$801$	10365.7	0.457244
$802$	0	0
$803$	25641.9	1.12688
$804$	0	0
$805$	1026.18	0.0449291
$806$	0	0
$807$	20939.9	0.913406
$808$	0	0
$809$	10119.6	0.439787	0.219894	−	0.975524i	$-0.429429\pi$
$809$	10119.6	0.439787	0.219894	+	0.975524i	$0.429429\pi$
$810$	0	0
$811$	−23079.6	−0.999303	−0.499652	−	0.866226i	$-0.666539\pi$
$811$	−23079.6	−0.999303	−0.499652	+	0.866226i	$0.666539\pi$
$812$	0	0
$813$	−54412.7	−2.34728
$814$	0	0
$815$	−2118.84	−0.0910672
$816$	0	0
$817$	−79.8385	−0.00341884
$818$	0	0
$819$	17560.1	0.749207
$820$	0	0
$821$	10608.1	0.450945	0.225473	−	0.974250i	$-0.427607\pi$
$821$	10608.1	0.450945	0.225473	+	0.974250i	$0.427607\pi$
$822$	0	0
$823$	6114.71	0.258986	0.129493	−	0.991580i	$-0.458665\pi$
$823$	6114.71	0.258986	0.129493	+	0.991580i	$0.458665\pi$
$824$	0	0
$825$	−27399.1	−1.15626
$826$	0	0
$827$	31584.8	1.32807	0.664033	−	0.747703i	$-0.268843\pi$
$827$	31584.8	1.32807	0.664033	+	0.747703i	$0.268843\pi$
$828$	0	0
$829$	24207.2	1.01417	0.507087	−	0.861895i	$-0.330722\pi$
$829$	24207.2	1.01417	0.507087	+	0.861895i	$0.330722\pi$
$830$	0	0
$831$	16311.7	0.680921
$832$	0	0
$833$	45099.4	1.87587
$834$	0	0
$835$	−6497.01	−0.269267
$836$	0	0
$837$	6851.19	0.282929
$838$	0	0
$839$	38899.3	1.60066	0.800329	−	0.599561i	$-0.204658\pi$
$839$	38899.3	1.60066	0.800329	+	0.599561i	$0.204658\pi$
$840$	0	0
$841$	841.000	0.0344828
$842$	0	0
$843$	50830.0	2.07672
$844$	0	0
$845$	457.344	0.0186191
$846$	0	0
$847$	−3605.51	−0.146265
$848$	0	0
$849$	48029.3	1.94153
$850$	0	0
$851$	−4816.68	−0.194023
$852$	0	0
$853$	−39006.0	−1.56570	−0.782848	−	0.622213i	$-0.786234\pi$
$853$	−39006.0	−1.56570	−0.782848	+	0.622213i	$0.786234\pi$
$854$	0	0
$855$	1057.26	0.0422896
$856$	0	0
$857$	−11708.7	−0.466700	−0.233350	−	0.972393i	$-0.574969\pi$
$857$	−11708.7	−0.466700	−0.233350	+	0.972393i	$0.574969\pi$
$858$	0	0
$859$	39476.8	1.56802	0.784011	−	0.620747i	$-0.213171\pi$
$859$	39476.8	1.56802	0.784011	+	0.620747i	$0.213171\pi$
$860$	0	0
$861$	−55748.6	−2.20663
$862$	0	0
$863$	26810.7	1.05753	0.528764	−	0.848769i	$-0.322656\pi$
$863$	26810.7	1.05753	0.528764	+	0.848769i	$0.322656\pi$
$864$	0	0
$865$	−2182.22	−0.0857777
$866$	0	0
$867$	85222.7	3.33831
$868$	0	0
$869$	36774.0	1.43553
$870$	0	0
$871$	7034.95	0.273674
$872$	0	0
$873$	−1581.50	−0.0613125
$874$	0	0
$875$	11225.2	0.433694
$876$	0	0
$877$	−36718.7	−1.41380	−0.706899	−	0.707314i	$-0.749907\pi$
$877$	−36718.7	−1.41380	−0.706899	+	0.707314i	$0.749907\pi$
$878$	0	0
$879$	−31726.5	−1.21742
$880$	0	0
$881$	−30253.6	−1.15695	−0.578473	−	0.815701i	$-0.696351\pi$
$881$	−30253.6	−1.15695	−0.578473	+	0.815701i	$0.696351\pi$
$882$	0	0
$883$	12267.1	0.467522	0.233761	−	0.972294i	$-0.424897\pi$
$883$	12267.1	0.467522	0.233761	+	0.972294i	$0.424897\pi$
$884$	0	0
$885$	3321.51	0.126160
$886$	0	0
$887$	45007.2	1.70371	0.851856	−	0.523776i	$-0.175477\pi$
$887$	45007.2	1.70371	0.851856	+	0.523776i	$0.175477\pi$
$888$	0	0
$889$	−46066.4	−1.73793
$890$	0	0
$891$	−31355.2	−1.17894
$892$	0	0
$893$	−647.526	−0.0242650
$894$	0	0
$895$	−5516.77	−0.206039
$896$	0	0
$897$	−6460.13	−0.240465
$898$	0	0
$899$	2614.90	0.0970099
$900$	0	0
$901$	20055.6	0.741564
$902$	0	0
$903$	342.092	0.0126070
$904$	0	0
$905$	5097.83	0.187246
$906$	0	0
$907$	17604.7	0.644494	0.322247	−	0.946656i	$-0.395562\pi$
$907$	17604.7	0.644494	0.322247	+	0.946656i	$0.395562\pi$
$908$	0	0
$909$	21392.3	0.780568
$910$	0	0
$911$	−32031.3	−1.16492	−0.582461	−	0.812859i	$-0.697910\pi$
$911$	−32031.3	−1.16492	−0.582461	+	0.812859i	$0.697910\pi$
$912$	0	0
$913$	−17152.1	−0.621742
$914$	0	0
$915$	1265.20	0.0457118
$916$	0	0
$917$	−47523.5	−1.71141
$918$	0	0
$919$	11659.6	0.418515	0.209257	−	0.977861i	$-0.432895\pi$
$919$	11659.6	0.418515	0.209257	+	0.977861i	$0.432895\pi$
$920$	0	0
$921$	−43731.1	−1.56459
$922$	0	0
$923$	5689.26	0.202886
$924$	0	0
$925$	−26020.0	−0.924901
$926$	0	0
$927$	28016.9	0.992658
$928$	0	0
$929$	25455.5	0.898997	0.449499	−	0.893281i	$-0.351603\pi$
$929$	25455.5	0.898997	0.449499	+	0.893281i	$0.351603\pi$
$930$	0	0
$931$	−13294.2	−0.467990
$932$	0	0
$933$	10955.2	0.384414
$934$	0	0
$935$	−8085.03	−0.282790
$936$	0	0
$937$	18209.7	0.634882	0.317441	−	0.948278i	$-0.397176\pi$
$937$	18209.7	0.634882	0.317441	+	0.948278i	$0.397176\pi$
$938$	0	0
$939$	−62112.4	−2.15864
$940$	0	0
$941$	−28483.5	−0.986755	−0.493377	−	0.869815i	$-0.664238\pi$
$941$	−28483.5	−0.986755	−0.493377	+	0.869815i	$0.664238\pi$
$942$	0	0
$943$	7424.45	0.256387
$944$	0	0
$945$	3453.69	0.118887
$946$	0	0
$947$	20823.1	0.714529	0.357264	−	0.934003i	$-0.383710\pi$
$947$	20823.1	0.714529	0.357264	+	0.934003i	$0.383710\pi$
$948$	0	0
$949$	32660.0	1.11716
$950$	0	0
$951$	−26543.0	−0.905063
$952$	0	0
$953$	−5514.74	−0.187450	−0.0937251	−	0.995598i	$-0.529877\pi$
$953$	−5514.74	−0.187450	−0.0937251	+	0.995598i	$0.529877\pi$
$954$	0	0
$955$	384.299	0.0130216
$956$	0	0
$957$	−6515.17	−0.220069
$958$	0	0
$959$	−62996.1	−2.12122
$960$	0	0
$961$	−21660.5	−0.727083
$962$	0	0
$963$	17110.5	0.572563
$964$	0	0
$965$	5681.12	0.189515
$966$	0	0
$967$	−42258.1	−1.40530	−0.702652	−	0.711534i	$-0.748001\pi$
$967$	−42258.1	−1.40530	−0.702652	+	0.711534i	$0.748001\pi$
$968$	0	0
$969$	−34542.9	−1.14518
$970$	0	0
$971$	−11158.8	−0.368798	−0.184399	−	0.982851i	$-0.559034\pi$
$971$	−11158.8	−0.368798	−0.184399	+	0.982851i	$0.559034\pi$
$972$	0	0
$973$	−46587.2	−1.53496
$974$	0	0
$975$	−34898.0	−1.14629
$976$	0	0
$977$	−32314.4	−1.05817	−0.529083	−	0.848570i	$-0.677464\pi$
$977$	−32314.4	−1.05817	−0.529083	+	0.848570i	$0.677464\pi$
$978$	0	0
$979$	23366.1	0.762802
$980$	0	0
$981$	−9166.11	−0.298319
$982$	0	0
$983$	18702.9	0.606847	0.303424	−	0.952856i	$-0.401870\pi$
$983$	18702.9	0.606847	0.303424	+	0.952856i	$0.401870\pi$
$984$	0	0
$985$	−6579.18	−0.212822
$986$	0	0
$987$	2774.51	0.0894769
$988$	0	0
$989$	−45.5589	−0.00146480
$990$	0	0
$991$	11042.5	0.353962	0.176981	−	0.984214i	$-0.443367\pi$
$991$	11042.5	0.353962	0.176981	+	0.984214i	$0.443367\pi$
$992$	0	0
$993$	13098.0	0.418584
$994$	0	0
$995$	4356.14	0.138793
$996$	0	0
$997$	−26609.4	−0.845265	−0.422633	−	0.906301i	$-0.638894\pi$
$997$	−26609.4	−0.845265	−0.422633	+	0.906301i	$0.638894\pi$
$998$	0	0
$999$	−16211.0	−0.513405

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	928.4.a.e.1.8	yes	9
4.3	odd	2		928.4.a.d.1.2	✓	9
8.3	odd	2		1856.4.a.bg.1.8		9
8.5	even	2		1856.4.a.bf.1.2		9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
928.4.a.d.1.2	✓	9	4.3	odd	2
928.4.a.e.1.8	yes	9	1.1	even	1	trivial
1856.4.a.bf.1.2		9	8.5	even	2
1856.4.a.bg.1.8		9	8.3	odd	2

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:	\( N \)	\(=\)	\( 928 = 2^{5} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	928.a (trivial)

\(f(q)\)	\(=\)	\(q+6.50540 q^{3} -1.74434 q^{5} +26.0581 q^{7} +15.3202 q^{9} +O(q^{10})\)	\(q+6.50540 q^{3} -1.74434 q^{5} +26.0581 q^{7} +15.3202 q^{9} +34.5346 q^{11} +43.9865 q^{13} -11.3476 q^{15} +134.214 q^{17} -39.5628 q^{19} +169.519 q^{21} -22.5760 q^{23} -121.957 q^{25} -75.9817 q^{27} -29.0000 q^{29} -90.1691 q^{31} +224.661 q^{33} -45.4542 q^{35} +213.354 q^{37} +286.150 q^{39} -328.864 q^{41} +2.01802 q^{43} -26.7236 q^{45} +16.3670 q^{47} +336.027 q^{49} +873.113 q^{51} +149.430 q^{53} -60.2400 q^{55} -257.372 q^{57} -292.706 q^{59} -111.495 q^{61} +399.216 q^{63} -76.7273 q^{65} +159.934 q^{67} -146.866 q^{69} +129.341 q^{71} +742.501 q^{73} -793.381 q^{75} +899.907 q^{77} +1064.85 q^{79} -907.937 q^{81} -496.664 q^{83} -234.114 q^{85} -188.657 q^{87} +676.600 q^{89} +1146.21 q^{91} -586.586 q^{93} +69.0110 q^{95} -103.230 q^{97} +529.077 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q + 4 q^{3} + 10 q^{5} + 12 q^{7} + 49 q^{9}+O(q^{10}) \) 9 * q + 4 * q^3 + 10 * q^5 + 12 * q^7 + 49 * q^9	\( 9 q + 4 q^{3} + 10 q^{5} + 12 q^{7} + 49 q^{9} + 64 q^{11} + 70 q^{13} + 170 q^{15} - 66 q^{17} + 42 q^{19} + 76 q^{21} + 40 q^{23} + 111 q^{25} + 322 q^{27} - 261 q^{29} - 64 q^{31} - 52 q^{33} + 496 q^{35} - 54 q^{37} + 590 q^{39} - 378 q^{41} - 32 q^{43} + 1046 q^{45} + 1164 q^{47} - 351 q^{49} + 376 q^{51} + 278 q^{53} + 614 q^{55} + 28 q^{57} + 640 q^{59} + 1054 q^{61} + 1660 q^{63} - 708 q^{65} + 1184 q^{67} + 188 q^{69} + 1988 q^{71} - 750 q^{73} + 3126 q^{75} + 1260 q^{77} + 2916 q^{79} + 293 q^{81} + 2832 q^{83} + 56 q^{85} - 116 q^{87} - 370 q^{89} + 3016 q^{91} - 1696 q^{93} + 4412 q^{95} - 2234 q^{97} + 4118 q^{99}+O(q^{100}) \) 9 * q + 4 * q^3 + 10 * q^5 + 12 * q^7 + 49 * q^9 + 64 * q^11 + 70 * q^13 + 170 * q^15 - 66 * q^17 + 42 * q^19 + 76 * q^21 + 40 * q^23 + 111 * q^25 + 322 * q^27 - 261 * q^29 - 64 * q^31 - 52 * q^33 + 496 * q^35 - 54 * q^37 + 590 * q^39 - 378 * q^41 - 32 * q^43 + 1046 * q^45 + 1164 * q^47 - 351 * q^49 + 376 * q^51 + 278 * q^53 + 614 * q^55 + 28 * q^57 + 640 * q^59 + 1054 * q^61 + 1660 * q^63 - 708 * q^65 + 1184 * q^67 + 188 * q^69 + 1988 * q^71 - 750 * q^73 + 3126 * q^75 + 1260 * q^77 + 2916 * q^79 + 293 * q^81 + 2832 * q^83 + 56 * q^85 - 116 * q^87 - 370 * q^89 + 3016 * q^91 - 1696 * q^93 + 4412 * q^95 - 2234 * q^97 + 4118 * q^99

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