LMFDB - Embedded newform 1856.4.a.u.1.2

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:	$3$
Coefficient field:	3.3.229.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 4x - 1 $ x^3 - 4*x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 232)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.11491$ 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)

$f(q)$	$=$	$q+2.92622 q^{3} -14.3315 q^{5} -16.8804 q^{7} -18.4372 q^{9} +O(q^{10})$	$q+2.92622 q^{3} -14.3315 q^{5} -16.8804 q^{7} -18.4372 q^{9} +16.5613 q^{11} -55.3675 q^{13} -41.9372 q^{15} +6.28120 q^{17} -119.008 q^{19} -49.3957 q^{21} -22.5433 q^{23} +80.3924 q^{25} -132.959 q^{27} +29.0000 q^{29} -228.987 q^{31} +48.4621 q^{33} +241.921 q^{35} -257.186 q^{37} -162.017 q^{39} +382.208 q^{41} -170.123 q^{43} +264.234 q^{45} +172.889 q^{47} -58.0532 q^{49} +18.3802 q^{51} +69.2381 q^{53} -237.349 q^{55} -348.244 q^{57} -43.6860 q^{59} -684.892 q^{61} +311.227 q^{63} +793.500 q^{65} -528.585 q^{67} -65.9667 q^{69} +488.054 q^{71} +80.3863 q^{73} +235.246 q^{75} -279.561 q^{77} +741.898 q^{79} +108.737 q^{81} -1360.29 q^{83} -90.0191 q^{85} +84.8604 q^{87} +957.563 q^{89} +934.623 q^{91} -670.068 q^{93} +1705.57 q^{95} -1111.59 q^{97} -305.345 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 6 q^{3} - 4 q^{5} - 16 q^{7} + 45 q^{9}+O(q^{10}) $ 3 * q + 6 * q^3 - 4 * q^5 - 16 * q^7 + 45 * q^9	$ 3 q + 6 q^{3} - 4 q^{5} - 16 q^{7} + 45 q^{9} - 2 q^{11} - 28 q^{13} - 136 q^{15} - 66 q^{17} - 66 q^{19} - 472 q^{21} - 176 q^{23} - 9 q^{25} + 228 q^{27} + 87 q^{29} + 190 q^{31} + 154 q^{33} + 660 q^{35} - 442 q^{37} + 656 q^{39} + 1162 q^{41} + 30 q^{43} + 254 q^{45} + 738 q^{47} + 851 q^{49} - 576 q^{51} - 312 q^{53} - 464 q^{55} + 684 q^{57} + 44 q^{59} - 54 q^{61} - 964 q^{63} + 178 q^{65} - 116 q^{67} - 812 q^{69} + 1200 q^{71} - 1118 q^{73} - 1038 q^{75} - 792 q^{77} + 2262 q^{79} + 15 q^{81} - 1804 q^{83} + 8 q^{85} + 174 q^{87} + 1578 q^{89} - 1972 q^{91} + 706 q^{93} + 1052 q^{95} + 1450 q^{97} - 482 q^{99}+O(q^{100}) $ 3 * q + 6 * q^3 - 4 * q^5 - 16 * q^7 + 45 * q^9 - 2 * q^11 - 28 * q^13 - 136 * q^15 - 66 * q^17 - 66 * q^19 - 472 * q^21 - 176 * q^23 - 9 * q^25 + 228 * q^27 + 87 * q^29 + 190 * q^31 + 154 * q^33 + 660 * q^35 - 442 * q^37 + 656 * q^39 + 1162 * q^41 + 30 * q^43 + 254 * q^45 + 738 * q^47 + 851 * q^49 - 576 * q^51 - 312 * q^53 - 464 * q^55 + 684 * q^57 + 44 * q^59 - 54 * q^61 - 964 * q^63 + 178 * q^65 - 116 * q^67 - 812 * q^69 + 1200 * q^71 - 1118 * q^73 - 1038 * q^75 - 792 * q^77 + 2262 * q^79 + 15 * q^81 - 1804 * q^83 + 8 * q^85 + 174 * q^87 + 1578 * q^89 - 1972 * q^91 + 706 * q^93 + 1052 * q^95 + 1450 * q^97 - 482 * 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
$3$	2.92622	0.563151	0.281576	−	0.959539i	$-0.409143\pi$
$3$	2.92622	0.563151	0.281576	+	0.959539i	$0.409143\pi$
$4$	0	0
$5$	−14.3315	−1.28185	−0.640925	−	0.767603i	$-0.721449\pi$
$5$	−14.3315	−1.28185	−0.640925	+	0.767603i	$0.721449\pi$
$6$	0	0
$7$	−16.8804	−0.911454	−0.455727	−	0.890120i	$-0.650621\pi$
$7$	−16.8804	−0.911454	−0.455727	+	0.890120i	$0.650621\pi$
$8$	0	0
$9$	−18.4372	−0.682860
$10$	0	0
$11$	16.5613	0.453948	0.226974	−	0.973901i	$-0.427117\pi$
$11$	16.5613	0.453948	0.226974	+	0.973901i	$0.427117\pi$
$12$	0	0
$13$	−55.3675	−1.18124	−0.590622	−	0.806948i	$-0.701118\pi$
$13$	−55.3675	−1.18124	−0.590622	+	0.806948i	$0.701118\pi$
$14$	0	0
$15$	−41.9372	−0.721876
$16$	0	0
$17$	6.28120	0.0896126	0.0448063	−	0.998996i	$-0.485733\pi$
$17$	6.28120	0.0896126	0.0448063	+	0.998996i	$0.485733\pi$
$18$	0	0
$19$	−119.008	−1.43696	−0.718482	−	0.695545i	$-0.755163\pi$
$19$	−119.008	−1.43696	−0.718482	+	0.695545i	$0.755163\pi$
$20$	0	0
$21$	−49.3957	−0.513287
$22$	0	0
$23$	−22.5433	−0.204374	−0.102187	−	0.994765i	$-0.532584\pi$
$23$	−22.5433	−0.204374	−0.102187	+	0.994765i	$0.532584\pi$
$24$	0	0
$25$	80.3924	0.643139
$26$	0	0
$27$	−132.959	−0.947705
$28$	0	0
$29$	29.0000	0.185695
$29$	29.0000	0.185695
$30$	0	0
$31$	−228.987	−1.32669	−0.663344	−	0.748314i	$-0.730864\pi$
$31$	−228.987	−1.32669	−0.663344	+	0.748314i	$0.730864\pi$
$32$	0	0
$33$	48.4621	0.255642
$34$	0	0
$35$	241.921	1.16835
$36$	0	0
$37$	−257.186	−1.14273	−0.571366	−	0.820696i	$-0.693586\pi$
$37$	−257.186	−1.14273	−0.571366	+	0.820696i	$0.693586\pi$
$38$	0	0
$39$	−162.017	−0.665219
$40$	0	0
$41$	382.208	1.45588	0.727938	−	0.685643i	$-0.240479\pi$
$41$	382.208	1.45588	0.727938	+	0.685643i	$0.240479\pi$
$42$	0	0
$43$	−170.123	−0.603337	−0.301669	−	0.953413i	$-0.597544\pi$
$43$	−170.123	−0.603337	−0.301669	+	0.953413i	$0.597544\pi$
$44$	0	0
$45$	264.234	0.875325
$46$	0	0
$47$	172.889	0.536564	0.268282	−	0.963340i	$-0.413544\pi$
$47$	172.889	0.536564	0.268282	+	0.963340i	$0.413544\pi$
$48$	0	0
$49$	−58.0532	−0.169251
$50$	0	0
$51$	18.3802	0.0504654
$52$	0	0
$53$	69.2381	0.179445	0.0897224	−	0.995967i	$-0.471402\pi$
$53$	69.2381	0.179445	0.0897224	+	0.995967i	$0.471402\pi$
$54$	0	0
$55$	−237.349	−0.581893
$56$	0	0
$57$	−348.244	−0.809229
$58$	0	0
$59$	−43.6860	−0.0963971	−0.0481986	−	0.998838i	$-0.515348\pi$
$59$	−43.6860	−0.0963971	−0.0481986	+	0.998838i	$0.515348\pi$
$60$	0	0
$61$	−684.892	−1.43756	−0.718782	−	0.695235i	$-0.755300\pi$
$61$	−684.892	−1.43756	−0.718782	+	0.695235i	$0.755300\pi$
$62$	0	0
$63$	311.227	0.622396
$64$	0	0
$65$	793.500	1.51418
$66$	0	0
$67$	−528.585	−0.963835	−0.481917	−	0.876217i	$-0.660059\pi$
$67$	−528.585	−0.963835	−0.481917	+	0.876217i	$0.660059\pi$
$68$	0	0
$69$	−65.9667	−0.115094
$70$	0	0
$71$	488.054	0.815794	0.407897	−	0.913028i	$-0.366262\pi$
$71$	488.054	0.815794	0.407897	+	0.913028i	$0.366262\pi$
$72$	0	0
$73$	80.3863	0.128884	0.0644418	−	0.997921i	$-0.479473\pi$
$73$	80.3863	0.128884	0.0644418	+	0.997921i	$0.479473\pi$
$74$	0	0
$75$	235.246	0.362185
$76$	0	0
$77$	−279.561	−0.413753
$78$	0	0
$79$	741.898	1.05658	0.528292	−	0.849063i	$-0.322833\pi$
$79$	741.898	1.05658	0.528292	+	0.849063i	$0.322833\pi$
$80$	0	0
$81$	108.737	0.149159
$82$	0	0
$83$	−1360.29	−1.79893	−0.899466	−	0.436991i	$-0.856044\pi$
$83$	−1360.29	−1.79893	−0.899466	+	0.436991i	$0.856044\pi$
$84$	0	0
$85$	−90.0191	−0.114870
$86$	0	0
$87$	84.8604	0.104575
$88$	0	0
$89$	957.563	1.14047	0.570233	−	0.821483i	$-0.306853\pi$
$89$	957.563	1.14047	0.570233	+	0.821483i	$0.306853\pi$
$90$	0	0
$91$	934.623	1.07665
$92$	0	0
$93$	−670.068	−0.747127
$94$	0	0
$95$	1705.57	1.84197
$96$	0	0
$97$	−1111.59	−1.16355	−0.581776	−	0.813349i	$-0.697642\pi$
$97$	−1111.59	−1.16355	−0.581776	+	0.813349i	$0.697642\pi$
$98$	0	0
$99$	−305.345	−0.309983
$100$	0	0
$101$	1691.56	1.66650	0.833249	−	0.552898i	$-0.186478\pi$
$101$	1691.56	1.66650	0.833249	+	0.552898i	$0.186478\pi$
$102$	0	0
$103$	439.312	0.420259	0.210130	−	0.977674i	$-0.432611\pi$
$103$	439.312	0.420259	0.210130	+	0.977674i	$0.432611\pi$
$104$	0	0
$105$	707.915	0.657957
$106$	0	0
$107$	−861.531	−0.778387	−0.389193	−	0.921156i	$-0.627246\pi$
$107$	−861.531	−0.778387	−0.389193	+	0.921156i	$0.627246\pi$
$108$	0	0
$109$	459.780	0.404027	0.202013	−	0.979383i	$-0.435252\pi$
$109$	459.780	0.404027	0.202013	+	0.979383i	$0.435252\pi$
$110$	0	0
$111$	−752.582	−0.643531
$112$	0	0
$113$	−9.56230	−0.00796058	−0.00398029	−	0.999992i	$-0.501267\pi$
$113$	−9.56230	−0.00796058	−0.00398029	+	0.999992i	$0.501267\pi$
$114$	0	0
$115$	323.080	0.261977
$116$	0	0
$117$	1020.82	0.806625
$118$	0	0
$119$	−106.029	−0.0816777
$120$	0	0
$121$	−1056.72	−0.793931
$122$	0	0
$123$	1118.43	0.819879
$124$	0	0
$125$	639.294	0.457442
$126$	0	0
$127$	−905.522	−0.632694	−0.316347	−	0.948644i	$-0.602456\pi$
$127$	−905.522	−0.632694	−0.316347	+	0.948644i	$0.602456\pi$
$128$	0	0
$129$	−497.817	−0.339770
$130$	0	0
$131$	1039.43	0.693250	0.346625	−	0.938004i	$-0.387328\pi$
$131$	1039.43	0.693250	0.346625	+	0.938004i	$0.387328\pi$
$132$	0	0
$133$	2008.90	1.30973
$134$	0	0
$135$	1905.51	1.21482
$136$	0	0
$137$	1146.58	0.715028	0.357514	−	0.933908i	$-0.383624\pi$
$137$	1146.58	0.715028	0.357514	+	0.933908i	$0.383624\pi$
$138$	0	0
$139$	−573.043	−0.349675	−0.174838	−	0.984597i	$-0.555940\pi$
$139$	−573.043	−0.349675	−0.174838	+	0.984597i	$0.555940\pi$
$140$	0	0
$141$	505.912	0.302167
$142$	0	0
$143$	−916.959	−0.536224
$144$	0	0
$145$	−415.614	−0.238034
$146$	0	0
$147$	−169.877	−0.0953142
$148$	0	0
$149$	−2324.32	−1.27796	−0.638979	−	0.769224i	$-0.720643\pi$
$149$	−2324.32	−1.27796	−0.638979	+	0.769224i	$0.720643\pi$
$150$	0	0
$151$	3188.68	1.71848	0.859242	−	0.511570i	$-0.170936\pi$
$151$	3188.68	1.71848	0.859242	+	0.511570i	$0.170936\pi$
$152$	0	0
$153$	−115.808	−0.0611929
$154$	0	0
$155$	3281.74	1.70062
$156$	0	0
$157$	2960.71	1.50503	0.752516	−	0.658574i	$-0.228840\pi$
$157$	2960.71	1.50503	0.752516	+	0.658574i	$0.228840\pi$
$158$	0	0
$159$	202.606	0.101055
$160$	0	0
$161$	380.540	0.186278
$162$	0	0
$163$	3417.14	1.64203	0.821015	−	0.570907i	$-0.193408\pi$
$163$	3417.14	1.64203	0.821015	+	0.570907i	$0.193408\pi$
$164$	0	0
$165$	−694.536	−0.327694
$166$	0	0
$167$	677.988	0.314158	0.157079	−	0.987586i	$-0.449792\pi$
$167$	677.988	0.314158	0.157079	+	0.987586i	$0.449792\pi$
$168$	0	0
$169$	868.556	0.395337
$170$	0	0
$171$	2194.18	0.981246
$172$	0	0
$173$	773.271	0.339831	0.169915	−	0.985459i	$-0.445651\pi$
$173$	773.271	0.339831	0.169915	+	0.985459i	$0.445651\pi$
$174$	0	0
$175$	−1357.05	−0.586192
$176$	0	0
$177$	−127.835	−0.0542862
$178$	0	0
$179$	1413.92	0.590399	0.295199	−	0.955436i	$-0.404614\pi$
$179$	1413.92	0.590399	0.295199	+	0.955436i	$0.404614\pi$
$180$	0	0
$181$	−2160.51	−0.887234	−0.443617	−	0.896216i	$-0.646305\pi$
$181$	−2160.51	−0.887234	−0.443617	+	0.896216i	$0.646305\pi$
$182$	0	0
$183$	−2004.15	−0.809567
$184$	0	0
$185$	3685.86	1.46481
$186$	0	0
$187$	104.025	0.0406795
$188$	0	0
$189$	2244.40	0.863790
$190$	0	0
$191$	−1586.39	−0.600981	−0.300490	−	0.953785i	$-0.597150\pi$
$191$	−1586.39	−0.600981	−0.300490	+	0.953785i	$0.597150\pi$
$192$	0	0
$193$	−1970.84	−0.735047	−0.367523	−	0.930014i	$-0.619794\pi$
$193$	−1970.84	−0.735047	−0.367523	+	0.930014i	$0.619794\pi$
$194$	0	0
$195$	2321.96	0.852711
$196$	0	0
$197$	−4803.57	−1.73726	−0.868629	−	0.495463i	$-0.834998\pi$
$197$	−4803.57	−1.73726	−0.868629	+	0.495463i	$0.834998\pi$
$198$	0	0
$199$	2692.85	0.959252	0.479626	−	0.877473i	$-0.340772\pi$
$199$	2692.85	0.959252	0.479626	+	0.877473i	$0.340772\pi$
$200$	0	0
$201$	−1546.76	−0.542785
$202$	0	0
$203$	−489.531	−0.169253
$204$	0	0
$205$	−5477.63	−1.86621
$206$	0	0
$207$	415.636	0.139559
$208$	0	0
$209$	−1970.93	−0.652308
$210$	0	0
$211$	−703.260	−0.229452	−0.114726	−	0.993397i	$-0.536599\pi$
$211$	−703.260	−0.229452	−0.114726	+	0.993397i	$0.536599\pi$
$212$	0	0
$213$	1428.15	0.459415
$214$	0	0
$215$	2438.12	0.773388
$216$	0	0
$217$	3865.39	1.20922
$218$	0	0
$219$	235.228	0.0725810
$220$	0	0
$221$	−347.774	−0.105854
$222$	0	0
$223$	3965.95	1.19094	0.595469	−	0.803378i	$-0.296966\pi$
$223$	3965.95	1.19094	0.595469	+	0.803378i	$0.296966\pi$
$224$	0	0
$225$	−1482.21	−0.439174
$226$	0	0
$227$	3976.09	1.16256	0.581282	−	0.813702i	$-0.302551\pi$
$227$	3976.09	1.16256	0.581282	+	0.813702i	$0.302551\pi$
$228$	0	0
$229$	611.004	0.176316	0.0881578	−	0.996107i	$-0.471902\pi$
$229$	611.004	0.176316	0.0881578	+	0.996107i	$0.471902\pi$
$230$	0	0
$231$	−818.058	−0.233006
$232$	0	0
$233$	−1051.90	−0.295761	−0.147880	−	0.989005i	$-0.547245\pi$
$233$	−1051.90	−0.295761	−0.147880	+	0.989005i	$0.547245\pi$
$234$	0	0
$235$	−2477.77	−0.687794
$236$	0	0
$237$	2170.96	0.595017
$238$	0	0
$239$	−2085.97	−0.564561	−0.282280	−	0.959332i	$-0.591091\pi$
$239$	−2085.97	−0.564561	−0.282280	+	0.959332i	$0.591091\pi$
$240$	0	0
$241$	−6064.55	−1.62096	−0.810481	−	0.585765i	$-0.800794\pi$
$241$	−6064.55	−1.62096	−0.810481	+	0.585765i	$0.800794\pi$
$242$	0	0
$243$	3908.09	1.03170
$244$	0	0
$245$	831.991	0.216955
$246$	0	0
$247$	6589.18	1.69741
$248$	0	0
$249$	−3980.51	−1.01307
$250$	0	0
$251$	5169.26	1.29992	0.649962	−	0.759967i	$-0.274785\pi$
$251$	5169.26	1.29992	0.649962	+	0.759967i	$0.274785\pi$
$252$	0	0
$253$	−373.347	−0.0927753
$254$	0	0
$255$	−263.416	−0.0646891
$256$	0	0
$257$	−7238.84	−1.75699	−0.878495	−	0.477751i	$-0.841452\pi$
$257$	−7238.84	−1.75699	−0.878495	+	0.477751i	$0.841452\pi$
$258$	0	0
$259$	4341.39	1.04155
$260$	0	0
$261$	−534.680	−0.126804
$262$	0	0
$263$	−4855.39	−1.13839	−0.569195	−	0.822203i	$-0.692745\pi$
$263$	−4855.39	−1.13839	−0.569195	+	0.822203i	$0.692745\pi$
$264$	0	0
$265$	−992.286	−0.230021
$266$	0	0
$267$	2802.04	0.642255
$268$	0	0
$269$	−5559.70	−1.26015	−0.630076	−	0.776533i	$-0.716976\pi$
$269$	−5559.70	−1.26015	−0.630076	+	0.776533i	$0.716976\pi$
$270$	0	0
$271$	−2534.36	−0.568087	−0.284044	−	0.958811i	$-0.591676\pi$
$271$	−2534.36	−0.568087	−0.284044	+	0.958811i	$0.591676\pi$
$272$	0	0
$273$	2734.91	0.606317
$274$	0	0
$275$	1331.41	0.291952
$276$	0	0
$277$	−7540.72	−1.63566	−0.817831	−	0.575459i	$-0.804823\pi$
$277$	−7540.72	−1.63566	−0.817831	+	0.575459i	$0.804823\pi$
$278$	0	0
$279$	4221.89	0.905943
$280$	0	0
$281$	5426.97	1.15212	0.576060	−	0.817407i	$-0.304589\pi$
$281$	5426.97	1.15212	0.576060	+	0.817407i	$0.304589\pi$
$282$	0	0
$283$	1297.75	0.272590	0.136295	−	0.990668i	$-0.456480\pi$
$283$	1297.75	0.272590	0.136295	+	0.990668i	$0.456480\pi$
$284$	0	0
$285$	4990.87	1.03731
$286$	0	0
$287$	−6451.82	−1.32696
$288$	0	0
$289$	−4873.55	−0.991970
$290$	0	0
$291$	−3252.75	−0.655256
$292$	0	0
$293$	6201.84	1.23657	0.618285	−	0.785954i	$-0.287828\pi$
$293$	6201.84	1.23657	0.618285	+	0.785954i	$0.287828\pi$
$294$	0	0
$295$	626.086	0.123567
$296$	0	0
$297$	−2201.98	−0.430209
$298$	0	0
$299$	1248.17	0.241416
$300$	0	0
$301$	2871.74	0.549914
$302$	0	0
$303$	4949.87	0.938491
$304$	0	0
$305$	9815.55	1.84274
$306$	0	0
$307$	−9181.06	−1.70681	−0.853406	−	0.521247i	$-0.825467\pi$
$307$	−9181.06	−1.70681	−0.853406	+	0.521247i	$0.825467\pi$
$308$	0	0
$309$	1285.52	0.236670
$310$	0	0
$311$	−3753.29	−0.684339	−0.342170	−	0.939638i	$-0.611162\pi$
$311$	−3753.29	−0.684339	−0.342170	+	0.939638i	$0.611162\pi$
$312$	0	0
$313$	2115.29	0.381991	0.190996	−	0.981591i	$-0.438828\pi$
$313$	2115.29	0.381991	0.190996	+	0.981591i	$0.438828\pi$
$314$	0	0
$315$	−4460.36	−0.797818
$316$	0	0
$317$	4739.89	0.839807	0.419903	−	0.907569i	$-0.362064\pi$
$317$	4739.89	0.839807	0.419903	+	0.907569i	$0.362064\pi$
$318$	0	0
$319$	480.279	0.0842961
$320$	0	0
$321$	−2521.03	−0.438350
$322$	0	0
$323$	−747.513	−0.128770
$324$	0	0
$325$	−4451.12	−0.759705
$326$	0	0
$327$	1345.42	0.227528
$328$	0	0
$329$	−2918.43	−0.489053
$330$	0	0
$331$	−6958.68	−1.15554	−0.577770	−	0.816200i	$-0.696077\pi$
$331$	−6958.68	−1.15554	−0.577770	+	0.816200i	$0.696077\pi$
$332$	0	0
$333$	4741.79	0.780326
$334$	0	0
$335$	7575.42	1.23549
$336$	0	0
$337$	2997.55	0.484530	0.242265	−	0.970210i	$-0.422110\pi$
$337$	2997.55	0.484530	0.242265	+	0.970210i	$0.422110\pi$
$338$	0	0
$339$	−27.9814	−0.00448301
$340$	0	0
$341$	−3792.34	−0.602248
$342$	0	0
$343$	6769.93	1.06572
$344$	0	0
$345$	945.404	0.147533
$346$	0	0
$347$	7788.46	1.20492	0.602459	−	0.798150i	$-0.294188\pi$
$347$	7788.46	1.20492	0.602459	+	0.798150i	$0.294188\pi$
$348$	0	0
$349$	6404.16	0.982254	0.491127	−	0.871088i	$-0.336585\pi$
$349$	6404.16	0.982254	0.491127	+	0.871088i	$0.336585\pi$
$350$	0	0
$351$	7361.62	1.11947
$352$	0	0
$353$	−5869.46	−0.884986	−0.442493	−	0.896772i	$-0.645906\pi$
$353$	−5869.46	−0.884986	−0.442493	+	0.896772i	$0.645906\pi$
$354$	0	0
$355$	−6994.55	−1.04572
$356$	0	0
$357$	−310.264	−0.0459969
$358$	0	0
$359$	2761.48	0.405975	0.202988	−	0.979181i	$-0.434935\pi$
$359$	2761.48	0.405975	0.202988	+	0.979181i	$0.434935\pi$
$360$	0	0
$361$	7303.93	1.06487
$362$	0	0
$363$	−3092.20	−0.447103
$364$	0	0
$365$	−1152.06	−0.165209
$366$	0	0
$367$	−3039.36	−0.432298	−0.216149	−	0.976360i	$-0.569350\pi$
$367$	−3039.36	−0.432298	−0.216149	+	0.976360i	$0.569350\pi$
$368$	0	0
$369$	−7046.86	−0.994160
$370$	0	0
$371$	−1168.76	−0.163556
$372$	0	0
$373$	−2051.57	−0.284789	−0.142394	−	0.989810i	$-0.545480\pi$
$373$	−2051.57	−0.284789	−0.142394	+	0.989810i	$0.545480\pi$
$374$	0	0
$375$	1870.72	0.257609
$376$	0	0
$377$	−1605.66	−0.219351
$378$	0	0
$379$	−1852.42	−0.251062	−0.125531	−	0.992090i	$-0.540063\pi$
$379$	−1852.42	−0.251062	−0.125531	+	0.992090i	$0.540063\pi$
$380$	0	0
$381$	−2649.76	−0.356302
$382$	0	0
$383$	−5723.43	−0.763587	−0.381793	−	0.924248i	$-0.624693\pi$
$383$	−5723.43	−0.763587	−0.381793	+	0.924248i	$0.624693\pi$
$384$	0	0
$385$	4006.54	0.530369
$386$	0	0
$387$	3136.59	0.411995
$388$	0	0
$389$	4946.70	0.644750	0.322375	−	0.946612i	$-0.395519\pi$
$389$	4946.70	0.644750	0.322375	+	0.946612i	$0.395519\pi$
$390$	0	0
$391$	−141.599	−0.0183145
$392$	0	0
$393$	3041.61	0.390405
$394$	0	0
$395$	−10632.5	−1.35438
$396$	0	0
$397$	−9275.78	−1.17264	−0.586320	−	0.810080i	$-0.699424\pi$
$397$	−9275.78	−1.17264	−0.586320	+	0.810080i	$0.699424\pi$
$398$	0	0
$399$	5878.49	0.737575
$400$	0	0
$401$	5051.01	0.629016	0.314508	−	0.949255i	$-0.398161\pi$
$401$	5051.01	0.629016	0.314508	+	0.949255i	$0.398161\pi$
$402$	0	0
$403$	12678.4	1.56714
$404$	0	0
$405$	−1558.36	−0.191199
$406$	0	0
$407$	−4259.34	−0.518741
$408$	0	0
$409$	5049.42	0.610459	0.305229	−	0.952279i	$-0.401267\pi$
$409$	5049.42	0.610459	0.305229	+	0.952279i	$0.401267\pi$
$410$	0	0
$411$	3355.14	0.402669
$412$	0	0
$413$	737.435	0.0878616
$414$	0	0
$415$	19495.0	2.30596
$416$	0	0
$417$	−1676.85	−0.196920
$418$	0	0
$419$	5332.54	0.621746	0.310873	−	0.950451i	$-0.399379\pi$
$419$	5332.54	0.621746	0.310873	+	0.950451i	$0.399379\pi$
$420$	0	0
$421$	−6195.89	−0.717266	−0.358633	−	0.933479i	$-0.616757\pi$
$421$	−6195.89	−0.717266	−0.358633	+	0.933479i	$0.616757\pi$
$422$	0	0
$423$	−3187.60	−0.366398
$424$	0	0
$425$	504.961	0.0576334
$426$	0	0
$427$	11561.2	1.31027
$428$	0	0
$429$	−2683.22	−0.301975
$430$	0	0
$431$	−1399.50	−0.156408	−0.0782038	−	0.996937i	$-0.524918\pi$
$431$	−1399.50	−0.156408	−0.0782038	+	0.996937i	$0.524918\pi$
$432$	0	0
$433$	1689.45	0.187505	0.0937526	−	0.995596i	$-0.470114\pi$
$433$	1689.45	0.187505	0.0937526	+	0.995596i	$0.470114\pi$
$434$	0	0
$435$	−1216.18	−0.134049
$436$	0	0
$437$	2682.84	0.293679
$438$	0	0
$439$	12352.8	1.34298	0.671491	−	0.741013i	$-0.265654\pi$
$439$	12352.8	1.34298	0.671491	+	0.741013i	$0.265654\pi$
$440$	0	0
$441$	1070.34	0.115575
$442$	0	0
$443$	−13427.1	−1.44005	−0.720023	−	0.693950i	$-0.755869\pi$
$443$	−13427.1	−1.44005	−0.720023	+	0.693950i	$0.755869\pi$
$444$	0	0
$445$	−13723.3	−1.46191
$446$	0	0
$447$	−6801.47	−0.719684
$448$	0	0
$449$	2912.34	0.306107	0.153053	−	0.988218i	$-0.451089\pi$
$449$	2912.34	0.306107	0.153053	+	0.988218i	$0.451089\pi$
$450$	0	0
$451$	6329.88	0.660892
$452$	0	0
$453$	9330.78	0.967767
$454$	0	0
$455$	−13394.6	−1.38010
$456$	0	0
$457$	5906.26	0.604558	0.302279	−	0.953219i	$-0.402253\pi$
$457$	5906.26	0.604558	0.302279	+	0.953219i	$0.402253\pi$
$458$	0	0
$459$	−835.144	−0.0849263
$460$	0	0
$461$	−2847.28	−0.287659	−0.143830	−	0.989602i	$-0.545942\pi$
$461$	−2847.28	−0.287659	−0.143830	+	0.989602i	$0.545942\pi$
$462$	0	0
$463$	−8561.02	−0.859319	−0.429659	−	0.902991i	$-0.641366\pi$
$463$	−8561.02	−0.859319	−0.429659	+	0.902991i	$0.641366\pi$
$464$	0	0
$465$	9603.09	0.957704
$466$	0	0
$467$	14998.9	1.48622	0.743112	−	0.669167i	$-0.233349\pi$
$467$	14998.9	1.48622	0.743112	+	0.669167i	$0.233349\pi$
$468$	0	0
$469$	8922.71	0.878491
$470$	0	0
$471$	8663.68	0.847561
$472$	0	0
$473$	−2817.46	−0.273884
$474$	0	0
$475$	−9567.35	−0.924169
$476$	0	0
$477$	−1276.56	−0.122536
$478$	0	0
$479$	8422.27	0.803389	0.401695	−	0.915774i	$-0.368421\pi$
$479$	8422.27	0.803389	0.401695	+	0.915774i	$0.368421\pi$
$480$	0	0
$481$	14239.7	1.34984
$482$	0	0
$483$	1113.54	0.104903
$484$	0	0
$485$	15930.7	1.49150
$486$	0	0
$487$	1773.91	0.165058	0.0825291	−	0.996589i	$-0.473700\pi$
$487$	1773.91	0.165058	0.0825291	+	0.996589i	$0.473700\pi$
$488$	0	0
$489$	9999.30	0.924711
$490$	0	0
$491$	−2157.20	−0.198275	−0.0991375	−	0.995074i	$-0.531608\pi$
$491$	−2157.20	−0.198275	−0.0991375	+	0.995074i	$0.531608\pi$
$492$	0	0
$493$	182.155	0.0166406
$494$	0	0
$495$	4376.06	0.397352
$496$	0	0
$497$	−8238.53	−0.743558
$498$	0	0
$499$	4303.31	0.386058	0.193029	−	0.981193i	$-0.438169\pi$
$499$	4303.31	0.386058	0.193029	+	0.981193i	$0.438169\pi$
$500$	0	0
$501$	1983.94	0.176918
$502$	0	0
$503$	−16020.5	−1.42011	−0.710057	−	0.704144i	$-0.751331\pi$
$503$	−16020.5	−1.42011	−0.710057	+	0.704144i	$0.751331\pi$
$504$	0	0
$505$	−24242.6	−2.13620
$506$	0	0
$507$	2541.59	0.222635
$508$	0	0
$509$	−18269.6	−1.59094	−0.795469	−	0.605994i	$-0.792776\pi$
$509$	−18269.6	−1.59094	−0.795469	+	0.605994i	$0.792776\pi$
$510$	0	0
$511$	−1356.95	−0.117472
$512$	0	0
$513$	15823.2	1.36182
$514$	0	0
$515$	−6296.01	−0.538709
$516$	0	0
$517$	2863.28	0.243572
$518$	0	0
$519$	2262.76	0.191376
$520$	0	0
$521$	13128.7	1.10399	0.551996	−	0.833847i	$-0.313866\pi$
$521$	13128.7	1.10399	0.551996	+	0.833847i	$0.313866\pi$
$522$	0	0
$523$	8295.67	0.693583	0.346792	−	0.937942i	$-0.387271\pi$
$523$	8295.67	0.693583	0.346792	+	0.937942i	$0.387271\pi$
$524$	0	0
$525$	−3971.04	−0.330115
$526$	0	0
$527$	−1438.31	−0.118888
$528$	0	0
$529$	−11658.8	−0.958231
$530$	0	0
$531$	805.449	0.0658258
$532$	0	0
$533$	−21161.9	−1.71974
$534$	0	0
$535$	12347.1	0.997775
$536$	0	0
$537$	4137.44	0.332484
$538$	0	0
$539$	−961.439	−0.0768314
$540$	0	0
$541$	−23563.6	−1.87261	−0.936303	−	0.351194i	$-0.885776\pi$
$541$	−23563.6	−1.87261	−0.936303	+	0.351194i	$0.885776\pi$
$542$	0	0
$543$	−6322.13	−0.499647
$544$	0	0
$545$	−6589.34	−0.517902
$546$	0	0
$547$	17151.4	1.34066	0.670330	−	0.742063i	$-0.266152\pi$
$547$	17151.4	1.34066	0.670330	+	0.742063i	$0.266152\pi$
$548$	0	0
$549$	12627.5	0.981656
$550$	0	0
$551$	−3451.23	−0.266838
$552$	0	0
$553$	−12523.5	−0.963027
$554$	0	0
$555$	10785.6	0.824910
$556$	0	0
$557$	−18572.6	−1.41283	−0.706415	−	0.707798i	$-0.749689\pi$
$557$	−18572.6	−1.41283	−0.706415	+	0.707798i	$0.749689\pi$
$558$	0	0
$559$	9419.27	0.712688
$560$	0	0
$561$	304.400	0.0229087
$562$	0	0
$563$	7835.40	0.586541	0.293270	−	0.956030i	$-0.405256\pi$
$563$	7835.40	0.586541	0.293270	+	0.956030i	$0.405256\pi$
$564$	0	0
$565$	137.042	0.0102043
$566$	0	0
$567$	−1835.52	−0.135951
$568$	0	0
$569$	5891.09	0.434038	0.217019	−	0.976167i	$-0.430367\pi$
$569$	5891.09	0.434038	0.217019	+	0.976167i	$0.430367\pi$
$570$	0	0
$571$	−20083.8	−1.47195	−0.735973	−	0.677010i	$-0.763275\pi$
$571$	−20083.8	−1.47195	−0.735973	+	0.677010i	$0.763275\pi$
$572$	0	0
$573$	−4642.13	−0.338443
$574$	0	0
$575$	−1812.31	−0.131441
$576$	0	0
$577$	−21008.9	−1.51579	−0.757896	−	0.652375i	$-0.773773\pi$
$577$	−21008.9	−1.51579	−0.757896	+	0.652375i	$0.773773\pi$
$578$	0	0
$579$	−5767.11	−0.413943
$580$	0	0
$581$	22962.2	1.63964
$582$	0	0
$583$	1146.67	0.0814587
$584$	0	0
$585$	−14629.9	−1.03397
$586$	0	0
$587$	−20222.3	−1.42192	−0.710958	−	0.703235i	$-0.751738\pi$
$587$	−20222.3	−1.42192	−0.710958	+	0.703235i	$0.751738\pi$
$588$	0	0
$589$	27251.4	1.90640
$590$	0	0
$591$	−14056.3	−0.978339
$592$	0	0
$593$	18803.0	1.30210	0.651051	−	0.759034i	$-0.274328\pi$
$593$	18803.0	1.30210	0.651051	+	0.759034i	$0.274328\pi$
$594$	0	0
$595$	1519.55	0.104699
$596$	0	0
$597$	7879.87	0.540204
$598$	0	0
$599$	−10612.1	−0.723872	−0.361936	−	0.932203i	$-0.617884\pi$
$599$	−10612.1	−0.723872	−0.361936	+	0.932203i	$0.617884\pi$
$600$	0	0
$601$	27709.6	1.88070	0.940350	−	0.340210i	$-0.110498\pi$
$601$	27709.6	1.88070	0.940350	+	0.340210i	$0.110498\pi$
$602$	0	0
$603$	9745.64	0.658164
$604$	0	0
$605$	15144.4	1.01770
$606$	0	0
$607$	21945.0	1.46741	0.733707	−	0.679466i	$-0.237788\pi$
$607$	21945.0	1.46741	0.733707	+	0.679466i	$0.237788\pi$
$608$	0	0
$609$	−1432.47	−0.0953150
$610$	0	0
$611$	−9572.44	−0.633813
$612$	0	0
$613$	11777.0	0.775971	0.387985	−	0.921665i	$-0.373171\pi$
$613$	11777.0	0.775971	0.387985	+	0.921665i	$0.373171\pi$
$614$	0	0
$615$	−16028.7	−1.05096
$616$	0	0
$617$	−27161.1	−1.77223	−0.886115	−	0.463465i	$-0.846606\pi$
$617$	−27161.1	−1.77223	−0.886115	+	0.463465i	$0.846606\pi$
$618$	0	0
$619$	−10366.3	−0.673116	−0.336558	−	0.941663i	$-0.609263\pi$
$619$	−10366.3	−0.673116	−0.336558	+	0.941663i	$0.609263\pi$
$620$	0	0
$621$	2997.35	0.193687
$622$	0	0
$623$	−16164.0	−1.03948
$624$	0	0
$625$	−19211.1	−1.22951
$626$	0	0
$627$	−5767.38	−0.367348
$628$	0	0
$629$	−1615.43	−0.102403
$630$	0	0
$631$	11208.4	0.707132	0.353566	−	0.935410i	$-0.384969\pi$
$631$	11208.4	0.707132	0.353566	+	0.935410i	$0.384969\pi$
$632$	0	0
$633$	−2057.89	−0.129216
$634$	0	0
$635$	12977.5	0.811018
$636$	0	0
$637$	3214.26	0.199927
$638$	0	0
$639$	−8998.36	−0.557073
$640$	0	0
$641$	351.957	0.0216872	0.0108436	−	0.999941i	$-0.496548\pi$
$641$	351.957	0.0216872	0.0108436	+	0.999941i	$0.496548\pi$
$642$	0	0
$643$	27628.0	1.69447	0.847234	−	0.531220i	$-0.178266\pi$
$643$	27628.0	1.69447	0.847234	+	0.531220i	$0.178266\pi$
$644$	0	0
$645$	7134.48	0.435534
$646$	0	0
$647$	10858.0	0.659771	0.329886	−	0.944021i	$-0.392990\pi$
$647$	10858.0	0.659771	0.329886	+	0.944021i	$0.392990\pi$
$648$	0	0
$649$	−723.498	−0.0437593
$650$	0	0
$651$	11311.0	0.680972
$652$	0	0
$653$	9714.54	0.582174	0.291087	−	0.956697i	$-0.405983\pi$
$653$	9714.54	0.582174	0.291087	+	0.956697i	$0.405983\pi$
$654$	0	0
$655$	−14896.7	−0.888643
$656$	0	0
$657$	−1482.10	−0.0880095
$658$	0	0
$659$	1918.79	0.113422	0.0567111	−	0.998391i	$-0.481939\pi$
$659$	1918.79	0.113422	0.0567111	+	0.998391i	$0.481939\pi$
$660$	0	0
$661$	12106.3	0.712375	0.356188	−	0.934414i	$-0.384076\pi$
$661$	12106.3	0.712375	0.356188	+	0.934414i	$0.384076\pi$
$662$	0	0
$663$	−1017.66	−0.0596120
$664$	0	0
$665$	−28790.6	−1.67887
$666$	0	0
$667$	−653.756	−0.0379513
$668$	0	0
$669$	11605.2	0.670679
$670$	0	0
$671$	−11342.7	−0.652580
$672$	0	0
$673$	16348.0	0.936355	0.468178	−	0.883634i	$-0.344911\pi$
$673$	16348.0	0.936355	0.468178	+	0.883634i	$0.344911\pi$
$674$	0	0
$675$	−10688.9	−0.609507
$676$	0	0
$677$	19298.2	1.09555	0.547776	−	0.836625i	$-0.315475\pi$
$677$	19298.2	1.09555	0.547776	+	0.836625i	$0.315475\pi$
$678$	0	0
$679$	18764.0	1.06052
$680$	0	0
$681$	11634.9	0.654700
$682$	0	0
$683$	9938.38	0.556781	0.278390	−	0.960468i	$-0.410199\pi$
$683$	9938.38	0.556781	0.278390	+	0.960468i	$0.410199\pi$
$684$	0	0
$685$	−16432.2	−0.916558
$686$	0	0
$687$	1787.93	0.0992924
$688$	0	0
$689$	−3833.54	−0.211968
$690$	0	0
$691$	−18610.4	−1.02456	−0.512282	−	0.858817i	$-0.671200\pi$
$691$	−18610.4	−1.02456	−0.512282	+	0.858817i	$0.671200\pi$
$692$	0	0
$693$	5154.34	0.282536
$694$	0	0
$695$	8212.58	0.448231
$696$	0	0
$697$	2400.73	0.130465
$698$	0	0
$699$	−3078.09	−0.166558
$700$	0	0
$701$	−26877.8	−1.44816	−0.724080	−	0.689716i	$-0.757735\pi$
$701$	−26877.8	−1.44816	−0.724080	+	0.689716i	$0.757735\pi$
$702$	0	0
$703$	30607.2	1.64206
$704$	0	0
$705$	−7250.49	−0.387332
$706$	0	0
$707$	−28554.1	−1.51894
$708$	0	0
$709$	3141.28	0.166394	0.0831970	−	0.996533i	$-0.473487\pi$
$709$	3141.28	0.166394	0.0831970	+	0.996533i	$0.473487\pi$
$710$	0	0
$711$	−13678.6	−0.721499
$712$	0	0
$713$	5162.14	0.271141
$714$	0	0
$715$	13141.4	0.687358
$716$	0	0
$717$	−6104.00	−0.317933
$718$	0	0
$719$	−27198.6	−1.41076	−0.705379	−	0.708830i	$-0.749223\pi$
$719$	−27198.6	−1.41076	−0.705379	+	0.708830i	$0.749223\pi$
$720$	0	0
$721$	−7415.75	−0.383047
$722$	0	0
$723$	−17746.2	−0.912847
$724$	0	0
$725$	2331.38	0.119428
$726$	0	0
$727$	−15713.1	−0.801603	−0.400802	−	0.916165i	$-0.631268\pi$
$727$	−15713.1	−0.801603	−0.400802	+	0.916165i	$0.631268\pi$
$728$	0	0
$729$	8500.05	0.431847
$730$	0	0
$731$	−1068.58	−0.0540666
$732$	0	0
$733$	−10977.4	−0.553153	−0.276576	−	0.960992i	$-0.589200\pi$
$733$	−10977.4	−0.553153	−0.276576	+	0.960992i	$0.589200\pi$
$734$	0	0
$735$	2434.59	0.122178
$736$	0	0
$737$	−8754.07	−0.437531
$738$	0	0
$739$	12586.3	0.626515	0.313258	−	0.949668i	$-0.398580\pi$
$739$	12586.3	0.626515	0.313258	+	0.949668i	$0.398580\pi$
$740$	0	0
$741$	19281.4	0.955897
$742$	0	0
$743$	−21237.6	−1.04863	−0.524315	−	0.851524i	$-0.675679\pi$
$743$	−21237.6	−1.04863	−0.524315	+	0.851524i	$0.675679\pi$
$744$	0	0
$745$	33311.0	1.63815
$746$	0	0
$747$	25080.0	1.22842
$748$	0	0
$749$	14543.0	0.709464
$750$	0	0
$751$	−31114.2	−1.51181	−0.755907	−	0.654679i	$-0.772804\pi$
$751$	−31114.2	−1.51181	−0.755907	+	0.654679i	$0.772804\pi$
$752$	0	0
$753$	15126.4	0.732054
$754$	0	0
$755$	−45698.6	−2.20284
$756$	0	0
$757$	−12838.7	−0.616423	−0.308211	−	0.951318i	$-0.599730\pi$
$757$	−12838.7	−0.616423	−0.308211	+	0.951318i	$0.599730\pi$
$758$	0	0
$759$	−1092.50	−0.0522466
$760$	0	0
$761$	−11665.6	−0.555689	−0.277844	−	0.960626i	$-0.589620\pi$
$761$	−11665.6	−0.555689	−0.277844	+	0.960626i	$0.589620\pi$
$762$	0	0
$763$	−7761.25	−0.368252
$764$	0	0
$765$	1659.70	0.0784401
$766$	0	0
$767$	2418.78	0.113869
$768$	0	0
$769$	−33339.3	−1.56339	−0.781693	−	0.623663i	$-0.785644\pi$
$769$	−33339.3	−1.56339	−0.781693	+	0.623663i	$0.785644\pi$
$770$	0	0
$771$	−21182.4	−0.989452
$772$	0	0
$773$	17895.2	0.832660	0.416330	−	0.909214i	$-0.363316\pi$
$773$	17895.2	0.832660	0.416330	+	0.909214i	$0.363316\pi$
$774$	0	0
$775$	−18408.9	−0.853246
$776$	0	0
$777$	12703.9	0.586549
$778$	0	0
$779$	−45485.9	−2.09204
$780$	0	0
$781$	8082.82	0.370328
$782$	0	0
$783$	−3855.82	−0.175984
$784$	0	0
$785$	−42431.4	−1.92923
$786$	0	0
$787$	−15816.4	−0.716384	−0.358192	−	0.933648i	$-0.616607\pi$
$787$	−15816.4	−0.716384	−0.358192	+	0.933648i	$0.616607\pi$
$788$	0	0
$789$	−14208.0	−0.641086
$790$	0	0
$791$	161.415	0.00725570
$792$	0	0
$793$	37920.7	1.69811
$794$	0	0
$795$	−2903.65	−0.129537
$796$	0	0
$797$	−21146.2	−0.939822	−0.469911	−	0.882714i	$-0.655714\pi$
$797$	−21146.2	−0.939822	−0.469911	+	0.882714i	$0.655714\pi$
$798$	0	0
$799$	1085.95	0.0480828
$800$	0	0
$801$	−17654.8	−0.778779
$802$	0	0
$803$	1331.30	0.0585065
$804$	0	0
$805$	−5453.71	−0.238780
$806$	0	0
$807$	−16268.9	−0.709657
$808$	0	0
$809$	−19190.2	−0.833982	−0.416991	−	0.908911i	$-0.636915\pi$
$809$	−19190.2	−0.833982	−0.416991	+	0.908911i	$0.636915\pi$
$810$	0	0
$811$	−37890.9	−1.64060	−0.820302	−	0.571931i	$-0.806194\pi$
$811$	−37890.9	−1.64060	−0.820302	+	0.571931i	$0.806194\pi$
$812$	0	0
$813$	−7416.11	−0.319919
$814$	0	0
$815$	−48972.8	−2.10484
$816$	0	0
$817$	20246.0	0.866974
$818$	0	0
$819$	−17231.9	−0.735201
$820$	0	0
$821$	11006.2	0.467868	0.233934	−	0.972252i	$-0.424840\pi$
$821$	11006.2	0.467868	0.233934	+	0.972252i	$0.424840\pi$
$822$	0	0
$823$	−42361.2	−1.79419	−0.897094	−	0.441839i	$-0.854326\pi$
$823$	−42361.2	−1.79419	−0.897094	+	0.441839i	$0.854326\pi$
$824$	0	0
$825$	3895.99	0.164413
$826$	0	0
$827$	−1786.13	−0.0751028	−0.0375514	−	0.999295i	$-0.511956\pi$
$827$	−1786.13	−0.0751028	−0.0375514	+	0.999295i	$0.511956\pi$
$828$	0	0
$829$	−12070.8	−0.505712	−0.252856	−	0.967504i	$-0.581370\pi$
$829$	−12070.8	−0.505712	−0.252856	+	0.967504i	$0.581370\pi$
$830$	0	0
$831$	−22065.8	−0.921125
$832$	0	0
$833$	−364.644	−0.0151671
$834$	0	0
$835$	−9716.60	−0.402703
$836$	0	0
$837$	30446.0	1.25731
$838$	0	0
$839$	39687.9	1.63311	0.816555	−	0.577267i	$-0.195881\pi$
$839$	39687.9	1.63311	0.816555	+	0.577267i	$0.195881\pi$
$840$	0	0
$841$	841.000	0.0344828
$842$	0	0
$843$	15880.5	0.648818
$844$	0	0
$845$	−12447.7	−0.506763
$846$	0	0
$847$	17837.9	0.723632
$848$	0	0
$849$	3797.50	0.153510
$850$	0	0
$851$	5797.82	0.233545
$852$	0	0
$853$	−36146.8	−1.45093	−0.725466	−	0.688258i	$-0.758376\pi$
$853$	−36146.8	−1.45093	−0.725466	+	0.688258i	$0.758376\pi$
$854$	0	0
$855$	−31445.9	−1.25781
$856$	0	0
$857$	1778.90	0.0709054	0.0354527	−	0.999371i	$-0.488713\pi$
$857$	1778.90	0.0709054	0.0354527	+	0.999371i	$0.488713\pi$
$858$	0	0
$859$	37890.5	1.50501	0.752507	−	0.658584i	$-0.228844\pi$
$859$	37890.5	1.50501	0.752507	+	0.658584i	$0.228844\pi$
$860$	0	0
$861$	−18879.4	−0.747282
$862$	0	0
$863$	22754.4	0.897529	0.448764	−	0.893650i	$-0.351864\pi$
$863$	22754.4	0.897529	0.448764	+	0.893650i	$0.351864\pi$
$864$	0	0
$865$	−11082.1	−0.435612
$866$	0	0
$867$	−14261.1	−0.558629
$868$	0	0
$869$	12286.8	0.479634
$870$	0	0
$871$	29266.4	1.13852
$872$	0	0
$873$	20494.6	0.794543
$874$	0	0
$875$	−10791.5	−0.416937
$876$	0	0
$877$	−28543.9	−1.09904	−0.549521	−	0.835480i	$-0.685190\pi$
$877$	−28543.9	−1.09904	−0.549521	+	0.835480i	$0.685190\pi$
$878$	0	0
$879$	18147.9	0.696376
$880$	0	0
$881$	46848.4	1.79156	0.895779	−	0.444499i	$-0.146618\pi$
$881$	46848.4	1.79156	0.895779	+	0.444499i	$0.146618\pi$
$882$	0	0
$883$	7020.06	0.267547	0.133773	−	0.991012i	$-0.457291\pi$
$883$	7020.06	0.267547	0.133773	+	0.991012i	$0.457291\pi$
$884$	0	0
$885$	1832.07	0.0695867
$886$	0	0
$887$	−19200.5	−0.726820	−0.363410	−	0.931629i	$-0.618388\pi$
$887$	−19200.5	−0.726820	−0.363410	+	0.931629i	$0.618388\pi$
$888$	0	0
$889$	15285.5	0.576671
$890$	0	0
$891$	1800.83	0.0677104
$892$	0	0
$893$	−20575.2	−0.771023
$894$	0	0
$895$	−20263.6	−0.756803
$896$	0	0
$897$	3652.41	0.135954
$898$	0	0
$899$	−6640.63	−0.246360
$900$	0	0
$901$	434.898	0.0160805
$902$	0	0
$903$	8403.34	0.309685
$904$	0	0
$905$	30963.4	1.13730
$906$	0	0
$907$	−36098.6	−1.32154	−0.660768	−	0.750590i	$-0.729769\pi$
$907$	−36098.6	−1.32154	−0.660768	+	0.750590i	$0.729769\pi$
$908$	0	0
$909$	−31187.6	−1.13799
$910$	0	0
$911$	19193.8	0.698047	0.349023	−	0.937114i	$-0.386513\pi$
$911$	19193.8	0.698047	0.349023	+	0.937114i	$0.386513\pi$
$912$	0	0
$913$	−22528.2	−0.816622
$914$	0	0
$915$	28722.5	1.03774
$916$	0	0
$917$	−17546.0	−0.631866
$918$	0	0
$919$	−49242.2	−1.76752	−0.883760	−	0.467941i	$-0.844996\pi$
$919$	−49242.2	−1.76752	−0.883760	+	0.467941i	$0.844996\pi$
$920$	0	0
$921$	−26865.8	−0.961193
$922$	0	0
$923$	−27022.3	−0.963651
$924$	0	0
$925$	−20675.8	−0.734936
$926$	0	0
$927$	−8099.70	−0.286978
$928$	0	0
$929$	−43350.9	−1.53100	−0.765498	−	0.643438i	$-0.777507\pi$
$929$	−43350.9	−1.53100	−0.765498	+	0.643438i	$0.777507\pi$
$930$	0	0
$931$	6908.80	0.243208
$932$	0	0
$933$	−10983.0	−0.385387
$934$	0	0
$935$	−1490.84	−0.0521450
$936$	0	0
$937$	39167.8	1.36559	0.682794	−	0.730611i	$-0.260765\pi$
$937$	39167.8	1.36559	0.682794	+	0.730611i	$0.260765\pi$
$938$	0	0
$939$	6189.81	0.215119
$940$	0	0
$941$	13329.9	0.461789	0.230895	−	0.972979i	$-0.425835\pi$
$941$	13329.9	0.461789	0.230895	+	0.972979i	$0.425835\pi$
$942$	0	0
$943$	−8616.25	−0.297544
$944$	0	0
$945$	−32165.7	−1.10725
$946$	0	0
$947$	−19115.3	−0.655926	−0.327963	−	0.944690i	$-0.606362\pi$
$947$	−19115.3	−0.655926	−0.327963	+	0.944690i	$0.606362\pi$
$948$	0	0
$949$	−4450.78	−0.152243
$950$	0	0
$951$	13870.0	0.472938
$952$	0	0
$953$	−29008.4	−0.986018	−0.493009	−	0.870024i	$-0.664103\pi$
$953$	−29008.4	−0.986018	−0.493009	+	0.870024i	$0.664103\pi$
$954$	0	0
$955$	22735.4	0.770367
$956$	0	0
$957$	1405.40	0.0474715
$958$	0	0
$959$	−19354.7	−0.651715
$960$	0	0
$961$	22644.2	0.760103
$962$	0	0
$963$	15884.3	0.531529
$964$	0	0
$965$	28245.1	0.942220
$966$	0	0
$967$	−55491.5	−1.84538	−0.922691	−	0.385539i	$-0.874015\pi$
$967$	−55491.5	−1.84538	−0.922691	+	0.385539i	$0.874015\pi$
$968$	0	0
$969$	−2187.39	−0.0725171
$970$	0	0
$971$	39721.0	1.31278	0.656389	−	0.754422i	$-0.272083\pi$
$971$	39721.0	1.31278	0.656389	+	0.754422i	$0.272083\pi$
$972$	0	0
$973$	9673.18	0.318713
$974$	0	0
$975$	−13025.0	−0.427829
$976$	0	0
$977$	4352.89	0.142540	0.0712699	−	0.997457i	$-0.477295\pi$
$977$	4352.89	0.142540	0.0712699	+	0.997457i	$0.477295\pi$
$978$	0	0
$979$	15858.5	0.517712
$980$	0	0
$981$	−8477.07	−0.275894
$982$	0	0
$983$	10317.1	0.334754	0.167377	−	0.985893i	$-0.446470\pi$
$983$	10317.1	0.334754	0.167377	+	0.985893i	$0.446470\pi$
$984$	0	0
$985$	68842.4	2.22690
$986$	0	0
$987$	−8539.98	−0.275411
$988$	0	0
$989$	3835.13	0.123307
$990$	0	0
$991$	45439.0	1.45653	0.728264	−	0.685297i	$-0.240328\pi$
$991$	45439.0	1.45653	0.728264	+	0.685297i	$0.240328\pi$
$992$	0	0
$993$	−20362.6	−0.650744
$994$	0	0
$995$	−38592.6	−1.22962
$996$	0	0
$997$	−31888.0	−1.01294	−0.506471	−	0.862257i	$-0.669050\pi$
$997$	−31888.0	−1.01294	−0.506471	+	0.862257i	$0.669050\pi$
$998$	0	0
$999$	34195.2	1.08297

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1856.4.a.u.1.2		3
4.3	odd	2		1856.4.a.p.1.2		3
8.3	odd	2		232.4.a.b.1.2	✓	3
8.5	even	2		464.4.a.g.1.2		3
24.11	even	2		2088.4.a.b.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
232.4.a.b.1.2	✓	3	8.3	odd	2
464.4.a.g.1.2		3	8.5	even	2
1856.4.a.p.1.2		3	4.3	odd	2
1856.4.a.u.1.2		3	1.1	even	1	trivial
2088.4.a.b.1.1		3	24.11	even	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:	\( N \)	\(=\)	\( 1856 = 2^{6} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1856.a (trivial)

\(f(q)\)	\(=\)	\(q+2.92622 q^{3} -14.3315 q^{5} -16.8804 q^{7} -18.4372 q^{9} +O(q^{10})\)	\(q+2.92622 q^{3} -14.3315 q^{5} -16.8804 q^{7} -18.4372 q^{9} +16.5613 q^{11} -55.3675 q^{13} -41.9372 q^{15} +6.28120 q^{17} -119.008 q^{19} -49.3957 q^{21} -22.5433 q^{23} +80.3924 q^{25} -132.959 q^{27} +29.0000 q^{29} -228.987 q^{31} +48.4621 q^{33} +241.921 q^{35} -257.186 q^{37} -162.017 q^{39} +382.208 q^{41} -170.123 q^{43} +264.234 q^{45} +172.889 q^{47} -58.0532 q^{49} +18.3802 q^{51} +69.2381 q^{53} -237.349 q^{55} -348.244 q^{57} -43.6860 q^{59} -684.892 q^{61} +311.227 q^{63} +793.500 q^{65} -528.585 q^{67} -65.9667 q^{69} +488.054 q^{71} +80.3863 q^{73} +235.246 q^{75} -279.561 q^{77} +741.898 q^{79} +108.737 q^{81} -1360.29 q^{83} -90.0191 q^{85} +84.8604 q^{87} +957.563 q^{89} +934.623 q^{91} -670.068 q^{93} +1705.57 q^{95} -1111.59 q^{97} -305.345 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 6 q^{3} - 4 q^{5} - 16 q^{7} + 45 q^{9}+O(q^{10}) \) 3 * q + 6 * q^3 - 4 * q^5 - 16 * q^7 + 45 * q^9	\( 3 q + 6 q^{3} - 4 q^{5} - 16 q^{7} + 45 q^{9} - 2 q^{11} - 28 q^{13} - 136 q^{15} - 66 q^{17} - 66 q^{19} - 472 q^{21} - 176 q^{23} - 9 q^{25} + 228 q^{27} + 87 q^{29} + 190 q^{31} + 154 q^{33} + 660 q^{35} - 442 q^{37} + 656 q^{39} + 1162 q^{41} + 30 q^{43} + 254 q^{45} + 738 q^{47} + 851 q^{49} - 576 q^{51} - 312 q^{53} - 464 q^{55} + 684 q^{57} + 44 q^{59} - 54 q^{61} - 964 q^{63} + 178 q^{65} - 116 q^{67} - 812 q^{69} + 1200 q^{71} - 1118 q^{73} - 1038 q^{75} - 792 q^{77} + 2262 q^{79} + 15 q^{81} - 1804 q^{83} + 8 q^{85} + 174 q^{87} + 1578 q^{89} - 1972 q^{91} + 706 q^{93} + 1052 q^{95} + 1450 q^{97} - 482 q^{99}+O(q^{100}) \) 3 * q + 6 * q^3 - 4 * q^5 - 16 * q^7 + 45 * q^9 - 2 * q^11 - 28 * q^13 - 136 * q^15 - 66 * q^17 - 66 * q^19 - 472 * q^21 - 176 * q^23 - 9 * q^25 + 228 * q^27 + 87 * q^29 + 190 * q^31 + 154 * q^33 + 660 * q^35 - 442 * q^37 + 656 * q^39 + 1162 * q^41 + 30 * q^43 + 254 * q^45 + 738 * q^47 + 851 * q^49 - 576 * q^51 - 312 * q^53 - 464 * q^55 + 684 * q^57 + 44 * q^59 - 54 * q^61 - 964 * q^63 + 178 * q^65 - 116 * q^67 - 812 * q^69 + 1200 * q^71 - 1118 * q^73 - 1038 * q^75 - 792 * q^77 + 2262 * q^79 + 15 * q^81 - 1804 * q^83 + 8 * q^85 + 174 * q^87 + 1578 * q^89 - 1972 * q^91 + 706 * q^93 + 1052 * q^95 + 1450 * q^97 - 482 * q^99

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