LMFDB - Embedded newform 1856.4.a.q.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("1856.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

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

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$109.507544971$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.4481.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 17x - 8 $ x^3 - 17*x - 8
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 232)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$4.34085$ 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)

$f(q)$	$=$	$q+7.68170 q^{3} -18.6859 q^{5} -10.3225 q^{7} +32.0084 q^{9} +O(q^{10})$	$q+7.68170 q^{3} -18.6859 q^{5} -10.3225 q^{7} +32.0084 q^{9} +17.0535 q^{11} +27.4043 q^{13} -143.540 q^{15} +48.9507 q^{17} -51.9098 q^{19} -79.2945 q^{21} +16.3958 q^{23} +224.163 q^{25} +38.4733 q^{27} +29.0000 q^{29} +240.920 q^{31} +131.000 q^{33} +192.886 q^{35} -300.378 q^{37} +210.511 q^{39} -105.531 q^{41} -519.203 q^{43} -598.107 q^{45} +375.092 q^{47} -236.445 q^{49} +376.024 q^{51} -673.993 q^{53} -318.661 q^{55} -398.755 q^{57} +195.559 q^{59} +636.965 q^{61} -330.408 q^{63} -512.074 q^{65} +81.8170 q^{67} +125.948 q^{69} -1004.51 q^{71} +137.343 q^{73} +1721.96 q^{75} -176.035 q^{77} +1198.46 q^{79} -568.688 q^{81} +137.252 q^{83} -914.689 q^{85} +222.769 q^{87} -204.708 q^{89} -282.881 q^{91} +1850.67 q^{93} +969.983 q^{95} +471.435 q^{97} +545.857 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{3} - 11 q^{5} - 38 q^{7} + 58 q^{9}+O(q^{10}) $ 3 * q - 3 * q^3 - 11 * q^5 - 38 * q^7 + 58 * q^9	$ 3 q - 3 q^{3} - 11 q^{5} - 38 q^{7} + 58 q^{9} - 65 q^{11} - 29 q^{13} - 85 q^{15} + 244 q^{17} - 312 q^{19} + 214 q^{21} - 24 q^{23} + 436 q^{25} - 57 q^{27} + 87 q^{29} + 249 q^{31} + 393 q^{33} + 718 q^{35} - 200 q^{37} + 749 q^{39} - 470 q^{41} - 97 q^{43} - 1562 q^{45} + 377 q^{47} + 383 q^{49} - 612 q^{51} - 1007 q^{53} - 1399 q^{55} + 1128 q^{57} + 364 q^{59} + 524 q^{61} - 2244 q^{63} + 301 q^{65} + 820 q^{67} + 824 q^{69} - 782 q^{71} + 1620 q^{73} + 1356 q^{75} - 158 q^{77} + 427 q^{79} + 227 q^{81} - 1520 q^{83} + 68 q^{85} - 87 q^{87} - 2474 q^{89} + 2002 q^{91} + 1807 q^{93} + 568 q^{95} - 170 q^{97} + 1362 q^{99}+O(q^{100}) $ 3 * q - 3 * q^3 - 11 * q^5 - 38 * q^7 + 58 * q^9 - 65 * q^11 - 29 * q^13 - 85 * q^15 + 244 * q^17 - 312 * q^19 + 214 * q^21 - 24 * q^23 + 436 * q^25 - 57 * q^27 + 87 * q^29 + 249 * q^31 + 393 * q^33 + 718 * q^35 - 200 * q^37 + 749 * q^39 - 470 * q^41 - 97 * q^43 - 1562 * q^45 + 377 * q^47 + 383 * q^49 - 612 * q^51 - 1007 * q^53 - 1399 * q^55 + 1128 * q^57 + 364 * q^59 + 524 * q^61 - 2244 * q^63 + 301 * q^65 + 820 * q^67 + 824 * q^69 - 782 * q^71 + 1620 * q^73 + 1356 * q^75 - 158 * q^77 + 427 * q^79 + 227 * q^81 - 1520 * q^83 + 68 * q^85 - 87 * q^87 - 2474 * q^89 + 2002 * q^91 + 1807 * q^93 + 568 * q^95 - 170 * q^97 + 1362 * 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
$3$	7.68170	1.47834	0.739171	−	0.673517i	$-0.235217\pi$
$3$	7.68170	1.47834	0.739171	+	0.673517i	$0.235217\pi$
$4$	0	0
$5$	−18.6859	−1.67132	−0.835660	−	0.549248i	$-0.814914\pi$
$5$	−18.6859	−1.67132	−0.835660	+	0.549248i	$0.814914\pi$
$6$	0	0
$7$	−10.3225	−0.557364	−0.278682	−	0.960383i	$-0.589898\pi$
$7$	−10.3225	−0.557364	−0.278682	+	0.960383i	$0.589898\pi$
$8$	0	0
$9$	32.0084	1.18550
$10$	0	0
$11$	17.0535	0.467439	0.233720	−	0.972304i	$-0.424910\pi$
$11$	17.0535	0.467439	0.233720	+	0.972304i	$0.424910\pi$
$12$	0	0
$13$	27.4043	0.584659	0.292330	−	0.956318i	$-0.405570\pi$
$13$	27.4043	0.584659	0.292330	+	0.956318i	$0.405570\pi$
$14$	0	0
$15$	−143.540	−2.47078
$16$	0	0
$17$	48.9507	0.698370	0.349185	−	0.937054i	$-0.386459\pi$
$17$	48.9507	0.698370	0.349185	+	0.937054i	$0.386459\pi$
$18$	0	0
$19$	−51.9098	−0.626786	−0.313393	−	0.949624i	$-0.601466\pi$
$19$	−51.9098	−0.626786	−0.313393	+	0.949624i	$0.601466\pi$
$20$	0	0
$21$	−79.2945	−0.823975
$22$	0	0
$23$	16.3958	0.148642	0.0743209	−	0.997234i	$-0.476321\pi$
$23$	16.3958	0.148642	0.0743209	+	0.997234i	$0.476321\pi$
$24$	0	0
$25$	224.163	1.79331
$26$	0	0
$27$	38.4733	0.274229
$28$	0	0
$29$	29.0000	0.185695
$29$	29.0000	0.185695
$30$	0	0
$31$	240.920	1.39582	0.697911	−	0.716185i	$-0.254113\pi$
$31$	240.920	1.39582	0.697911	+	0.716185i	$0.254113\pi$
$32$	0	0
$33$	131.000	0.691036
$34$	0	0
$35$	192.886	0.931533
$36$	0	0
$37$	−300.378	−1.33464	−0.667321	−	0.744770i	$-0.732559\pi$
$37$	−300.378	−1.33464	−0.667321	+	0.744770i	$0.732559\pi$
$38$	0	0
$39$	210.511	0.864327
$40$	0	0
$41$	−105.531	−0.401979	−0.200990	−	0.979593i	$-0.564416\pi$
$41$	−105.531	−0.401979	−0.200990	+	0.979593i	$0.564416\pi$
$42$	0	0
$43$	−519.203	−1.84134	−0.920671	−	0.390339i	$-0.872358\pi$
$43$	−519.203	−1.84134	−0.920671	+	0.390339i	$0.872358\pi$
$44$	0	0
$45$	−598.107	−1.98135
$46$	0	0
$47$	375.092	1.16410	0.582050	−	0.813153i	$-0.302251\pi$
$47$	375.092	1.16410	0.582050	+	0.813153i	$0.302251\pi$
$48$	0	0
$49$	−236.445	−0.689345
$50$	0	0
$51$	376.024	1.03243
$52$	0	0
$53$	−673.993	−1.74679	−0.873397	−	0.487009i	$-0.838088\pi$
$53$	−673.993	−1.74679	−0.873397	+	0.487009i	$0.838088\pi$
$54$	0	0
$55$	−318.661	−0.781240
$56$	0	0
$57$	−398.755	−0.926604
$58$	0	0
$59$	195.559	0.431519	0.215759	−	0.976447i	$-0.430777\pi$
$59$	195.559	0.431519	0.215759	+	0.976447i	$0.430777\pi$
$60$	0	0
$61$	636.965	1.33697	0.668484	−	0.743727i	$-0.266944\pi$
$61$	636.965	1.33697	0.668484	+	0.743727i	$0.266944\pi$
$62$	0	0
$63$	−330.408	−0.660754
$64$	0	0
$65$	−512.074	−0.977153
$66$	0	0
$67$	81.8170	0.149187	0.0745935	−	0.997214i	$-0.476234\pi$
$67$	81.8170	0.149187	0.0745935	+	0.997214i	$0.476234\pi$
$68$	0	0
$69$	125.948	0.219744
$70$	0	0
$71$	−1004.51	−1.67907	−0.839534	−	0.543308i	$-0.817172\pi$
$71$	−1004.51	−1.67907	−0.839534	+	0.543308i	$0.817172\pi$
$72$	0	0
$73$	137.343	0.220203	0.110102	−	0.993920i	$-0.464882\pi$
$73$	137.343	0.220203	0.110102	+	0.993920i	$0.464882\pi$
$74$	0	0
$75$	1721.96	2.65112
$76$	0	0
$77$	−176.035	−0.260534
$78$	0	0
$79$	1198.46	1.70680	0.853401	−	0.521254i	$-0.174536\pi$
$79$	1198.46	1.70680	0.853401	+	0.521254i	$0.174536\pi$
$80$	0	0
$81$	−568.688	−0.780093
$82$	0	0
$83$	137.252	0.181511	0.0907554	−	0.995873i	$-0.471072\pi$
$83$	137.252	0.181511	0.0907554	+	0.995873i	$0.471072\pi$
$84$	0	0
$85$	−914.689	−1.16720
$86$	0	0
$87$	222.769	0.274521
$88$	0	0
$89$	−204.708	−0.243809	−0.121905	−	0.992542i	$-0.538900\pi$
$89$	−204.708	−0.243809	−0.121905	+	0.992542i	$0.538900\pi$
$90$	0	0
$91$	−282.881	−0.325868
$92$	0	0
$93$	1850.67	2.06350
$94$	0	0
$95$	969.983	1.04756
$96$	0	0
$97$	471.435	0.493474	0.246737	−	0.969082i	$-0.420642\pi$
$97$	471.435	0.493474	0.246737	+	0.969082i	$0.420642\pi$
$98$	0	0
$99$	545.857	0.554148
$100$	0	0
$101$	−1662.49	−1.63786	−0.818929	−	0.573895i	$-0.805432\pi$
$101$	−1662.49	−1.63786	−0.818929	+	0.573895i	$0.805432\pi$
$102$	0	0
$103$	−702.730	−0.672253	−0.336126	−	0.941817i	$-0.609117\pi$
$103$	−702.730	−0.672253	−0.336126	+	0.941817i	$0.609117\pi$
$104$	0	0
$105$	1481.69	1.37713
$106$	0	0
$107$	−722.520	−0.652791	−0.326395	−	0.945233i	$-0.605834\pi$
$107$	−722.520	−0.652791	−0.326395	+	0.945233i	$0.605834\pi$
$108$	0	0
$109$	−1480.14	−1.30066	−0.650328	−	0.759654i	$-0.725368\pi$
$109$	−1480.14	−1.30066	−0.650328	+	0.759654i	$0.725368\pi$
$110$	0	0
$111$	−2307.41	−1.97306
$112$	0	0
$113$	1124.84	0.936425	0.468212	−	0.883616i	$-0.344898\pi$
$113$	1124.84	0.936425	0.468212	+	0.883616i	$0.344898\pi$
$114$	0	0
$115$	−306.371	−0.248428
$116$	0	0
$117$	877.168	0.693113
$118$	0	0
$119$	−505.295	−0.389246
$120$	0	0
$121$	−1040.18	−0.781501
$122$	0	0
$123$	−810.656	−0.594263
$124$	0	0
$125$	−1852.96	−1.32587
$126$	0	0
$127$	−2330.45	−1.62830	−0.814151	−	0.580653i	$-0.802797\pi$
$127$	−2330.45	−1.62830	−0.814151	+	0.580653i	$0.802797\pi$
$128$	0	0
$129$	−3988.36	−2.72214
$130$	0	0
$131$	1055.80	0.704166	0.352083	−	0.935969i	$-0.385474\pi$
$131$	1055.80	0.704166	0.352083	+	0.935969i	$0.385474\pi$
$132$	0	0
$133$	535.841	0.349348
$134$	0	0
$135$	−718.909	−0.458325
$136$	0	0
$137$	−2473.01	−1.54222	−0.771108	−	0.636705i	$-0.780297\pi$
$137$	−2473.01	−1.54222	−0.771108	+	0.636705i	$0.780297\pi$
$138$	0	0
$139$	−975.965	−0.595541	−0.297771	−	0.954637i	$-0.596243\pi$
$139$	−975.965	−0.595541	−0.297771	+	0.954637i	$0.596243\pi$
$140$	0	0
$141$	2881.34	1.72094
$142$	0	0
$143$	467.339	0.273293
$144$	0	0
$145$	−541.892	−0.310356
$146$	0	0
$147$	−1816.30	−1.01909
$148$	0	0
$149$	−1872.40	−1.02948	−0.514742	−	0.857345i	$-0.672112\pi$
$149$	−1872.40	−1.02948	−0.514742	+	0.857345i	$0.672112\pi$
$150$	0	0
$151$	315.404	0.169982	0.0849908	−	0.996382i	$-0.472914\pi$
$151$	315.404	0.169982	0.0849908	+	0.996382i	$0.472914\pi$
$152$	0	0
$153$	1566.84	0.827916
$154$	0	0
$155$	−4501.81	−2.33286
$156$	0	0
$157$	−1243.89	−0.632312	−0.316156	−	0.948707i	$-0.602392\pi$
$157$	−1243.89	−0.632312	−0.316156	+	0.948707i	$0.602392\pi$
$158$	0	0
$159$	−5177.41	−2.58236
$160$	0	0
$161$	−169.246	−0.0828476
$162$	0	0
$163$	721.053	0.346486	0.173243	−	0.984879i	$-0.444575\pi$
$163$	721.053	0.346486	0.173243	+	0.984879i	$0.444575\pi$
$164$	0	0
$165$	−2447.86	−1.15494
$166$	0	0
$167$	1017.41	0.471436	0.235718	−	0.971821i	$-0.424256\pi$
$167$	1017.41	0.471436	0.235718	+	0.971821i	$0.424256\pi$
$168$	0	0
$169$	−1446.01	−0.658173
$170$	0	0
$171$	−1661.55	−0.743053
$172$	0	0
$173$	1178.82	0.518056	0.259028	−	0.965870i	$-0.416598\pi$
$173$	1178.82	0.518056	0.259028	+	0.965870i	$0.416598\pi$
$174$	0	0
$175$	−2313.93	−0.999525
$176$	0	0
$177$	1502.22	0.637933
$178$	0	0
$179$	−1939.10	−0.809693	−0.404846	−	0.914385i	$-0.632675\pi$
$179$	−1939.10	−0.809693	−0.404846	+	0.914385i	$0.632675\pi$
$180$	0	0
$181$	1777.52	0.729955	0.364978	−	0.931016i	$-0.381077\pi$
$181$	1777.52	0.729955	0.364978	+	0.931016i	$0.381077\pi$
$182$	0	0
$183$	4892.97	1.97650
$184$	0	0
$185$	5612.83	2.23061
$186$	0	0
$187$	834.782	0.326445
$188$	0	0
$189$	−397.142	−0.152846
$190$	0	0
$191$	−3347.72	−1.26823	−0.634116	−	0.773238i	$-0.718636\pi$
$191$	−3347.72	−1.26823	−0.634116	+	0.773238i	$0.718636\pi$
$192$	0	0
$193$	−4108.04	−1.53214	−0.766070	−	0.642757i	$-0.777790\pi$
$193$	−4108.04	−1.53214	−0.766070	+	0.642757i	$0.777790\pi$
$194$	0	0
$195$	−3933.59	−1.44457
$196$	0	0
$197$	380.902	0.137757	0.0688785	−	0.997625i	$-0.478058\pi$
$197$	380.902	0.137757	0.0688785	+	0.997625i	$0.478058\pi$
$198$	0	0
$199$	−4665.50	−1.66195	−0.830975	−	0.556309i	$-0.812217\pi$
$199$	−4665.50	−1.66195	−0.830975	+	0.556309i	$0.812217\pi$
$200$	0	0
$201$	628.493	0.220550
$202$	0	0
$203$	−299.353	−0.103500
$204$	0	0
$205$	1971.94	0.671836
$206$	0	0
$207$	524.804	0.176215
$208$	0	0
$209$	−885.246	−0.292984
$210$	0	0
$211$	−6071.24	−1.98086	−0.990429	−	0.138021i	$-0.955926\pi$
$211$	−6071.24	−1.98086	−0.990429	+	0.138021i	$0.955926\pi$
$212$	0	0
$213$	−7716.36	−2.48224
$214$	0	0
$215$	9701.79	3.07747
$216$	0	0
$217$	−2486.90	−0.777981
$218$	0	0
$219$	1055.03	0.325536
$220$	0	0
$221$	1341.46	0.408309
$222$	0	0
$223$	1714.11	0.514731	0.257366	−	0.966314i	$-0.417146\pi$
$223$	1714.11	0.514731	0.257366	+	0.966314i	$0.417146\pi$
$224$	0	0
$225$	7175.12	2.12596
$226$	0	0
$227$	−2154.94	−0.630080	−0.315040	−	0.949078i	$-0.602018\pi$
$227$	−2154.94	−0.630080	−0.315040	+	0.949078i	$0.602018\pi$
$228$	0	0
$229$	6614.70	1.90878	0.954392	−	0.298557i	$-0.0965053\pi$
$229$	6614.70	1.90878	0.954392	+	0.298557i	$0.0965053\pi$
$230$	0	0
$231$	−1352.25	−0.385158
$232$	0	0
$233$	−1481.75	−0.416621	−0.208310	−	0.978063i	$-0.566796\pi$
$233$	−1481.75	−0.416621	−0.208310	+	0.978063i	$0.566796\pi$
$234$	0	0
$235$	−7008.93	−1.94558
$236$	0	0
$237$	9206.21	2.52324
$238$	0	0
$239$	1650.39	0.446674	0.223337	−	0.974741i	$-0.428305\pi$
$239$	1650.39	0.446674	0.223337	+	0.974741i	$0.428305\pi$
$240$	0	0
$241$	−751.384	−0.200834	−0.100417	−	0.994945i	$-0.532018\pi$
$241$	−751.384	−0.200834	−0.100417	+	0.994945i	$0.532018\pi$
$242$	0	0
$243$	−5407.26	−1.42747
$244$	0	0
$245$	4418.20	1.15212
$246$	0	0
$247$	−1422.55	−0.366456
$248$	0	0
$249$	1054.33	0.268335
$250$	0	0
$251$	138.745	0.0348904	0.0174452	−	0.999848i	$-0.494447\pi$
$251$	138.745	0.0348904	0.0174452	+	0.999848i	$0.494447\pi$
$252$	0	0
$253$	279.606	0.0694811
$254$	0	0
$255$	−7026.36	−1.72552
$256$	0	0
$257$	3281.85	0.796561	0.398280	−	0.917264i	$-0.369607\pi$
$257$	3281.85	0.796561	0.398280	+	0.917264i	$0.369607\pi$
$258$	0	0
$259$	3100.66	0.743882
$260$	0	0
$261$	928.245	0.220141
$262$	0	0
$263$	−226.941	−0.0532083	−0.0266042	−	0.999646i	$-0.508469\pi$
$263$	−226.941	−0.0532083	−0.0266042	+	0.999646i	$0.508469\pi$
$264$	0	0
$265$	12594.2	2.91945
$266$	0	0
$267$	−1572.51	−0.360434
$268$	0	0
$269$	532.087	0.120602	0.0603009	−	0.998180i	$-0.480794\pi$
$269$	532.087	0.120602	0.0603009	+	0.998180i	$0.480794\pi$
$270$	0	0
$271$	−362.895	−0.0813443	−0.0406721	−	0.999173i	$-0.512950\pi$
$271$	−362.895	−0.0813443	−0.0406721	+	0.999173i	$0.512950\pi$
$272$	0	0
$273$	−2173.01	−0.481745
$274$	0	0
$275$	3822.78	0.838262
$276$	0	0
$277$	6740.55	1.46210	0.731048	−	0.682326i	$-0.239032\pi$
$277$	6740.55	1.46210	0.731048	+	0.682326i	$0.239032\pi$
$278$	0	0
$279$	7711.47	1.65474
$280$	0	0
$281$	536.712	0.113942	0.0569708	−	0.998376i	$-0.481856\pi$
$281$	536.712	0.113942	0.0569708	+	0.998376i	$0.481856\pi$
$282$	0	0
$283$	−2022.13	−0.424746	−0.212373	−	0.977189i	$-0.568119\pi$
$283$	−2022.13	−0.424746	−0.212373	+	0.977189i	$0.568119\pi$
$284$	0	0
$285$	7451.11	1.54865
$286$	0	0
$287$	1089.35	0.224049
$288$	0	0
$289$	−2516.83	−0.512280
$290$	0	0
$291$	3621.42	0.729524
$292$	0	0
$293$	−8298.81	−1.65468	−0.827341	−	0.561700i	$-0.810148\pi$
$293$	−8298.81	−1.65468	−0.827341	+	0.561700i	$0.810148\pi$
$294$	0	0
$295$	−3654.20	−0.721205
$296$	0	0
$297$	656.106	0.128186
$298$	0	0
$299$	449.315	0.0869049
$300$	0	0
$301$	5359.49	1.02630
$302$	0	0
$303$	−12770.7	−2.42132
$304$	0	0
$305$	−11902.3	−2.23450
$306$	0	0
$307$	7404.94	1.37662	0.688310	−	0.725417i	$-0.258353\pi$
$307$	7404.94	1.37662	0.688310	+	0.725417i	$0.258353\pi$
$308$	0	0
$309$	−5398.16	−0.993820
$310$	0	0
$311$	7570.24	1.38029	0.690143	−	0.723673i	$-0.257548\pi$
$311$	7570.24	1.38029	0.690143	+	0.723673i	$0.257548\pi$
$312$	0	0
$313$	8227.37	1.48575	0.742873	−	0.669432i	$-0.233463\pi$
$313$	8227.37	1.48575	0.742873	+	0.669432i	$0.233463\pi$
$314$	0	0
$315$	6173.98	1.10433
$316$	0	0
$317$	1395.76	0.247298	0.123649	−	0.992326i	$-0.460540\pi$
$317$	1395.76	0.247298	0.123649	+	0.992326i	$0.460540\pi$
$318$	0	0
$319$	494.552	0.0868013
$320$	0	0
$321$	−5550.18	−0.965049
$322$	0	0
$323$	−2541.02	−0.437728
$324$	0	0
$325$	6143.03	1.04847
$326$	0	0
$327$	−11370.0	−1.92281
$328$	0	0
$329$	−3871.89	−0.648828
$330$	0	0
$331$	−1388.01	−0.230489	−0.115244	−	0.993337i	$-0.536765\pi$
$331$	−1388.01	−0.230489	−0.115244	+	0.993337i	$0.536765\pi$
$332$	0	0
$333$	−9614.62	−1.58222
$334$	0	0
$335$	−1528.83	−0.249339
$336$	0	0
$337$	9065.91	1.46544	0.732718	−	0.680533i	$-0.238252\pi$
$337$	9065.91	1.46544	0.732718	+	0.680533i	$0.238252\pi$
$338$	0	0
$339$	8640.67	1.38436
$340$	0	0
$341$	4108.53	0.652462
$342$	0	0
$343$	5981.34	0.941580
$344$	0	0
$345$	−2353.45	−0.367262
$346$	0	0
$347$	7460.96	1.15425	0.577126	−	0.816655i	$-0.304174\pi$
$347$	7460.96	1.15425	0.577126	+	0.816655i	$0.304174\pi$
$348$	0	0
$349$	−10312.0	−1.58162	−0.790812	−	0.612059i	$-0.790342\pi$
$349$	−10312.0	−1.58162	−0.790812	+	0.612059i	$0.790342\pi$
$350$	0	0
$351$	1054.33	0.160331
$352$	0	0
$353$	6146.73	0.926792	0.463396	−	0.886151i	$-0.346631\pi$
$353$	6146.73	0.926792	0.463396	+	0.886151i	$0.346631\pi$
$354$	0	0
$355$	18770.2	2.80626
$356$	0	0
$357$	−3881.52	−0.575439
$358$	0	0
$359$	647.637	0.0952116	0.0476058	−	0.998866i	$-0.484841\pi$
$359$	647.637	0.0952116	0.0476058	+	0.998866i	$0.484841\pi$
$360$	0	0
$361$	−4164.37	−0.607139
$362$	0	0
$363$	−7990.32	−1.15533
$364$	0	0
$365$	−2566.39	−0.368030
$366$	0	0
$367$	−11986.9	−1.70493	−0.852464	−	0.522785i	$-0.824893\pi$
$367$	−11986.9	−1.70493	−0.852464	+	0.522785i	$0.824893\pi$
$368$	0	0
$369$	−3377.88	−0.476546
$370$	0	0
$371$	6957.31	0.973600
$372$	0	0
$373$	−3929.25	−0.545439	−0.272720	−	0.962094i	$-0.587923\pi$
$373$	−3929.25	−0.545439	−0.272720	+	0.962094i	$0.587923\pi$
$374$	0	0
$375$	−14233.9	−1.96009
$376$	0	0
$377$	794.723	0.108569
$378$	0	0
$379$	5870.74	0.795672	0.397836	−	0.917456i	$-0.369761\pi$
$379$	5870.74	0.795672	0.397836	+	0.917456i	$0.369761\pi$
$380$	0	0
$381$	−17901.8	−2.40719
$382$	0	0
$383$	10112.1	1.34910	0.674549	−	0.738230i	$-0.264338\pi$
$383$	10112.1	1.34910	0.674549	+	0.738230i	$0.264338\pi$
$384$	0	0
$385$	3289.38	0.435435
$386$	0	0
$387$	−16618.9	−2.18291
$388$	0	0
$389$	11442.6	1.49143	0.745714	−	0.666266i	$-0.232109\pi$
$389$	11442.6	1.49143	0.745714	+	0.666266i	$0.232109\pi$
$390$	0	0
$391$	802.586	0.103807
$392$	0	0
$393$	8110.34	1.04100
$394$	0	0
$395$	−22394.3	−2.85261
$396$	0	0
$397$	−8776.12	−1.10947	−0.554737	−	0.832026i	$-0.687181\pi$
$397$	−8776.12	−1.10947	−0.554737	+	0.832026i	$0.687181\pi$
$398$	0	0
$399$	4116.16	0.516456
$400$	0	0
$401$	−8559.29	−1.06591	−0.532956	−	0.846143i	$-0.678919\pi$
$401$	−8559.29	−1.06591	−0.532956	+	0.846143i	$0.678919\pi$
$402$	0	0
$403$	6602.23	0.816080
$404$	0	0
$405$	10626.4	1.30378
$406$	0	0
$407$	−5122.50	−0.623864
$408$	0	0
$409$	14083.1	1.70261	0.851303	−	0.524674i	$-0.175813\pi$
$409$	14083.1	1.70261	0.851303	+	0.524674i	$0.175813\pi$
$410$	0	0
$411$	−18996.9	−2.27992
$412$	0	0
$413$	−2018.66	−0.240513
$414$	0	0
$415$	−2564.68	−0.303362
$416$	0	0
$417$	−7497.06	−0.880414
$418$	0	0
$419$	11355.4	1.32398	0.661992	−	0.749511i	$-0.269711\pi$
$419$	11355.4	1.32398	0.661992	+	0.749511i	$0.269711\pi$
$420$	0	0
$421$	−2532.89	−0.293220	−0.146610	−	0.989194i	$-0.546836\pi$
$421$	−2532.89	−0.293220	−0.146610	+	0.989194i	$0.546836\pi$
$422$	0	0
$423$	12006.1	1.38004
$424$	0	0
$425$	10973.0	1.25239
$426$	0	0
$427$	−6575.09	−0.745177
$428$	0	0
$429$	3589.96	0.404020
$430$	0	0
$431$	9175.84	1.02549	0.512743	−	0.858542i	$-0.328629\pi$
$431$	9175.84	1.02549	0.512743	+	0.858542i	$0.328629\pi$
$432$	0	0
$433$	−9707.88	−1.07744	−0.538719	−	0.842485i	$-0.681092\pi$
$433$	−9707.88	−1.07744	−0.538719	+	0.842485i	$0.681092\pi$
$434$	0	0
$435$	−4162.65	−0.458813
$436$	0	0
$437$	−851.104	−0.0931666
$438$	0	0
$439$	−13154.3	−1.43011	−0.715055	−	0.699068i	$-0.753598\pi$
$439$	−13154.3	−1.43011	−0.715055	+	0.699068i	$0.753598\pi$
$440$	0	0
$441$	−7568.25	−0.817217
$442$	0	0
$443$	−1546.92	−0.165906	−0.0829528	−	0.996553i	$-0.526435\pi$
$443$	−1546.92	−0.165906	−0.0829528	+	0.996553i	$0.526435\pi$
$444$	0	0
$445$	3825.16	0.407483
$446$	0	0
$447$	−14383.2	−1.52193
$448$	0	0
$449$	15412.5	1.61996	0.809981	−	0.586456i	$-0.199477\pi$
$449$	15412.5	1.61996	0.809981	+	0.586456i	$0.199477\pi$
$450$	0	0
$451$	−1799.67	−0.187901
$452$	0	0
$453$	2422.84	0.251291
$454$	0	0
$455$	5285.89	0.544630
$456$	0	0
$457$	294.671	0.0301622	0.0150811	−	0.999886i	$-0.495199\pi$
$457$	294.671	0.0301622	0.0150811	+	0.999886i	$0.495199\pi$
$458$	0	0
$459$	1883.30	0.191514
$460$	0	0
$461$	11942.0	1.20649	0.603246	−	0.797555i	$-0.293874\pi$
$461$	11942.0	1.20649	0.603246	+	0.797555i	$0.293874\pi$
$462$	0	0
$463$	−3668.22	−0.368200	−0.184100	−	0.982907i	$-0.558937\pi$
$463$	−3668.22	−0.368200	−0.184100	+	0.982907i	$0.558937\pi$
$464$	0	0
$465$	−34581.5	−3.44877
$466$	0	0
$467$	−4044.90	−0.400804	−0.200402	−	0.979714i	$-0.564225\pi$
$467$	−4044.90	−0.400804	−0.200402	+	0.979714i	$0.564225\pi$
$468$	0	0
$469$	−844.558	−0.0831515
$470$	0	0
$471$	−9555.16	−0.934774
$472$	0	0
$473$	−8854.24	−0.860716
$474$	0	0
$475$	−11636.3	−1.12402
$476$	0	0
$477$	−21573.5	−2.07082
$478$	0	0
$479$	−317.706	−0.0303055	−0.0151528	−	0.999885i	$-0.504823\pi$
$479$	−317.706	−0.0303055	−0.0151528	+	0.999885i	$0.504823\pi$
$480$	0	0
$481$	−8231.62	−0.780311
$482$	0	0
$483$	−1300.10	−0.122477
$484$	0	0
$485$	−8809.19	−0.824752
$486$	0	0
$487$	−17382.3	−1.61739	−0.808695	−	0.588228i	$-0.799826\pi$
$487$	−17382.3	−1.61739	−0.808695	+	0.588228i	$0.799826\pi$
$488$	0	0
$489$	5538.91	0.512225
$490$	0	0
$491$	−3884.83	−0.357067	−0.178534	−	0.983934i	$-0.557135\pi$
$491$	−3884.83	−0.357067	−0.178534	+	0.983934i	$0.557135\pi$
$492$	0	0
$493$	1419.57	0.129684
$494$	0	0
$495$	−10199.8	−0.926159
$496$	0	0
$497$	10369.1	0.935851
$498$	0	0
$499$	−9041.00	−0.811084	−0.405542	−	0.914076i	$-0.632917\pi$
$499$	−9041.00	−0.811084	−0.405542	+	0.914076i	$0.632917\pi$
$500$	0	0
$501$	7815.46	0.696945
$502$	0	0
$503$	12917.2	1.14503	0.572513	−	0.819896i	$-0.305969\pi$
$503$	12917.2	1.14503	0.572513	+	0.819896i	$0.305969\pi$
$504$	0	0
$505$	31065.1	2.73738
$506$	0	0
$507$	−11107.8	−0.973006
$508$	0	0
$509$	18731.5	1.63115	0.815577	−	0.578649i	$-0.196420\pi$
$509$	18731.5	1.63115	0.815577	+	0.578649i	$0.196420\pi$
$510$	0	0
$511$	−1417.73	−0.122733
$512$	0	0
$513$	−1997.14	−0.171883
$514$	0	0
$515$	13131.2	1.12355
$516$	0	0
$517$	6396.64	0.544147
$518$	0	0
$519$	9055.30	0.765864
$520$	0	0
$521$	11982.0	1.00757	0.503784	−	0.863830i	$-0.331941\pi$
$521$	11982.0	1.00757	0.503784	+	0.863830i	$0.331941\pi$
$522$	0	0
$523$	−12352.3	−1.03275	−0.516373	−	0.856363i	$-0.672718\pi$
$523$	−12352.3	−1.03275	−0.516373	+	0.856363i	$0.672718\pi$
$524$	0	0
$525$	−17774.9	−1.47764
$526$	0	0
$527$	11793.2	0.974800
$528$	0	0
$529$	−11898.2	−0.977906
$530$	0	0
$531$	6259.54	0.511564
$532$	0	0
$533$	−2892.00	−0.235021
$534$	0	0
$535$	13500.9	1.09102
$536$	0	0
$537$	−14895.6	−1.19700
$538$	0	0
$539$	−4032.23	−0.322227
$540$	0	0
$541$	−15520.5	−1.23342	−0.616710	−	0.787190i	$-0.711535\pi$
$541$	−15520.5	−1.23342	−0.616710	+	0.787190i	$0.711535\pi$
$542$	0	0
$543$	13654.4	1.07912
$544$	0	0
$545$	27657.7	2.17381
$546$	0	0
$547$	16581.6	1.29612	0.648061	−	0.761588i	$-0.275580\pi$
$547$	16581.6	1.29612	0.648061	+	0.761588i	$0.275580\pi$
$548$	0	0
$549$	20388.3	1.58497
$550$	0	0
$551$	−1505.39	−0.116391
$552$	0	0
$553$	−12371.1	−0.951310
$554$	0	0
$555$	43116.0	3.29761
$556$	0	0
$557$	−15760.6	−1.19892	−0.599459	−	0.800405i	$-0.704618\pi$
$557$	−15760.6	−1.19892	−0.599459	+	0.800405i	$0.704618\pi$
$558$	0	0
$559$	−14228.4	−1.07656
$560$	0	0
$561$	6412.54	0.482598
$562$	0	0
$563$	−18923.3	−1.41655	−0.708277	−	0.705934i	$-0.750527\pi$
$563$	−18923.3	−1.41655	−0.708277	+	0.705934i	$0.750527\pi$
$564$	0	0
$565$	−21018.7	−1.56506
$566$	0	0
$567$	5870.29	0.434796
$568$	0	0
$569$	−18641.2	−1.37343	−0.686715	−	0.726927i	$-0.740948\pi$
$569$	−18641.2	−1.37343	−0.686715	+	0.726927i	$0.740948\pi$
$570$	0	0
$571$	−13578.3	−0.995155	−0.497577	−	0.867420i	$-0.665777\pi$
$571$	−13578.3	−0.995155	−0.497577	+	0.867420i	$0.665777\pi$
$572$	0	0
$573$	−25716.1	−1.87488
$574$	0	0
$575$	3675.34	0.266561
$576$	0	0
$577$	23656.3	1.70680	0.853402	−	0.521254i	$-0.174536\pi$
$577$	23656.3	1.70680	0.853402	+	0.521254i	$0.174536\pi$
$578$	0	0
$579$	−31556.7	−2.26503
$580$	0	0
$581$	−1416.79	−0.101168
$582$	0	0
$583$	−11494.0	−0.816520
$584$	0	0
$585$	−16390.7	−1.15841
$586$	0	0
$587$	7671.11	0.539388	0.269694	−	0.962946i	$-0.413078\pi$
$587$	7671.11	0.539388	0.269694	+	0.962946i	$0.413078\pi$
$588$	0	0
$589$	−12506.1	−0.874881
$590$	0	0
$591$	2925.97	0.203652
$592$	0	0
$593$	−3200.46	−0.221631	−0.110815	−	0.993841i	$-0.535346\pi$
$593$	−3200.46	−0.221631	−0.110815	+	0.993841i	$0.535346\pi$
$594$	0	0
$595$	9441.90	0.650555
$596$	0	0
$597$	−35838.9	−2.45693
$598$	0	0
$599$	11062.9	0.754619	0.377309	−	0.926087i	$-0.376849\pi$
$599$	11062.9	0.754619	0.377309	+	0.926087i	$0.376849\pi$
$600$	0	0
$601$	−19979.9	−1.35607	−0.678035	−	0.735029i	$-0.737168\pi$
$601$	−19979.9	−1.35607	−0.678035	+	0.735029i	$0.737168\pi$
$602$	0	0
$603$	2618.83	0.176861
$604$	0	0
$605$	19436.7	1.30614
$606$	0	0
$607$	−9416.64	−0.629670	−0.314835	−	0.949146i	$-0.601949\pi$
$607$	−9416.64	−0.629670	−0.314835	+	0.949146i	$0.601949\pi$
$608$	0	0
$609$	−2299.54	−0.153008
$610$	0	0
$611$	10279.1	0.680603
$612$	0	0
$613$	8449.22	0.556706	0.278353	−	0.960479i	$-0.410211\pi$
$613$	8449.22	0.556706	0.278353	+	0.960479i	$0.410211\pi$
$614$	0	0
$615$	15147.9	0.993204
$616$	0	0
$617$	2679.89	0.174859	0.0874297	−	0.996171i	$-0.472135\pi$
$617$	2679.89	0.174859	0.0874297	+	0.996171i	$0.472135\pi$
$618$	0	0
$619$	6175.34	0.400982	0.200491	−	0.979696i	$-0.435746\pi$
$619$	6175.34	0.400982	0.200491	+	0.979696i	$0.435746\pi$
$620$	0	0
$621$	630.801	0.0407620
$622$	0	0
$623$	2113.11	0.135891
$624$	0	0
$625$	6603.82	0.422645
$626$	0	0
$627$	−6800.19	−0.433131
$628$	0	0
$629$	−14703.7	−0.932074
$630$	0	0
$631$	−19443.3	−1.22666	−0.613332	−	0.789825i	$-0.710171\pi$
$631$	−19443.3	−1.22666	−0.613332	+	0.789825i	$0.710171\pi$
$632$	0	0
$633$	−46637.4	−2.92839
$634$	0	0
$635$	43546.7	2.72141
$636$	0	0
$637$	−6479.61	−0.403032
$638$	0	0
$639$	−32152.9	−1.99053
$640$	0	0
$641$	−11202.8	−0.690305	−0.345153	−	0.938547i	$-0.612173\pi$
$641$	−11202.8	−0.690305	−0.345153	+	0.938547i	$0.612173\pi$
$642$	0	0
$643$	8365.65	0.513078	0.256539	−	0.966534i	$-0.417418\pi$
$643$	8365.65	0.513078	0.256539	+	0.966534i	$0.417418\pi$
$644$	0	0
$645$	74526.2	4.54956
$646$	0	0
$647$	3210.84	0.195102	0.0975511	−	0.995231i	$-0.468899\pi$
$647$	3210.84	0.195102	0.0975511	+	0.995231i	$0.468899\pi$
$648$	0	0
$649$	3334.97	0.201709
$650$	0	0
$651$	−19103.6	−1.15012
$652$	0	0
$653$	−11153.1	−0.668383	−0.334192	−	0.942505i	$-0.608463\pi$
$653$	−11153.1	−0.668383	−0.334192	+	0.942505i	$0.608463\pi$
$654$	0	0
$655$	−19728.6	−1.17689
$656$	0	0
$657$	4396.15	0.261050
$658$	0	0
$659$	12455.3	0.736253	0.368126	−	0.929776i	$-0.379999\pi$
$659$	12455.3	0.736253	0.368126	+	0.929776i	$0.379999\pi$
$660$	0	0
$661$	−8452.73	−0.497387	−0.248694	−	0.968582i	$-0.580001\pi$
$661$	−8452.73	−0.497387	−0.248694	+	0.968582i	$0.580001\pi$
$662$	0	0
$663$	10304.7	0.603620
$664$	0	0
$665$	−10012.7	−0.583872
$666$	0	0
$667$	475.479	0.0276021
$668$	0	0
$669$	13167.2	0.760949
$670$	0	0
$671$	10862.5	0.624951
$672$	0	0
$673$	22427.5	1.28457	0.642285	−	0.766466i	$-0.277987\pi$
$673$	22427.5	1.28457	0.642285	+	0.766466i	$0.277987\pi$
$674$	0	0
$675$	8624.31	0.491778
$676$	0	0
$677$	27328.8	1.55145	0.775725	−	0.631072i	$-0.217385\pi$
$677$	27328.8	1.55145	0.775725	+	0.631072i	$0.217385\pi$
$678$	0	0
$679$	−4866.40	−0.275045
$680$	0	0
$681$	−16553.6	−0.931474
$682$	0	0
$683$	10265.5	0.575106	0.287553	−	0.957765i	$-0.407158\pi$
$683$	10265.5	0.575106	0.287553	+	0.957765i	$0.407158\pi$
$684$	0	0
$685$	46210.5	2.57753
$686$	0	0
$687$	50812.1	2.82184
$688$	0	0
$689$	−18470.3	−1.02128
$690$	0	0
$691$	−24324.4	−1.33914	−0.669568	−	0.742751i	$-0.733521\pi$
$691$	−24324.4	−1.33914	−0.669568	+	0.742751i	$0.733521\pi$
$692$	0	0
$693$	−5634.62	−0.308862
$694$	0	0
$695$	18236.8	0.995340
$696$	0	0
$697$	−5165.81	−0.280730
$698$	0	0
$699$	−11382.4	−0.615908
$700$	0	0
$701$	−14240.5	−0.767270	−0.383635	−	0.923485i	$-0.625328\pi$
$701$	−14240.5	−0.767270	−0.383635	+	0.923485i	$0.625328\pi$
$702$	0	0
$703$	15592.5	0.836535
$704$	0	0
$705$	−53840.5	−2.87624
$706$	0	0
$707$	17161.1	0.912883
$708$	0	0
$709$	16127.5	0.854277	0.427138	−	0.904186i	$-0.359522\pi$
$709$	16127.5	0.854277	0.427138	+	0.904186i	$0.359522\pi$
$710$	0	0
$711$	38360.9	2.02341
$712$	0	0
$713$	3950.08	0.207478
$714$	0	0
$715$	−8732.66	−0.456759
$716$	0	0
$717$	12677.8	0.660337
$718$	0	0
$719$	−17816.7	−0.924129	−0.462065	−	0.886846i	$-0.652891\pi$
$719$	−17816.7	−0.924129	−0.462065	+	0.886846i	$0.652891\pi$
$720$	0	0
$721$	7253.95	0.374689
$722$	0	0
$723$	−5771.90	−0.296901
$724$	0	0
$725$	6500.74	0.333009
$726$	0	0
$727$	−25102.7	−1.28062	−0.640308	−	0.768118i	$-0.721193\pi$
$727$	−25102.7	−1.28062	−0.640308	+	0.768118i	$0.721193\pi$
$728$	0	0
$729$	−26182.4	−1.33020
$730$	0	0
$731$	−25415.4	−1.28594
$732$	0	0
$733$	13095.5	0.659882	0.329941	−	0.944002i	$-0.392971\pi$
$733$	13095.5	0.659882	0.329941	+	0.944002i	$0.392971\pi$
$734$	0	0
$735$	33939.3	1.70322
$736$	0	0
$737$	1395.27	0.0697359
$738$	0	0
$739$	16037.3	0.798297	0.399149	−	0.916886i	$-0.369306\pi$
$739$	16037.3	0.798297	0.399149	+	0.916886i	$0.369306\pi$
$740$	0	0
$741$	−10927.6	−0.541748
$742$	0	0
$743$	6959.89	0.343652	0.171826	−	0.985127i	$-0.445033\pi$
$743$	6959.89	0.343652	0.171826	+	0.985127i	$0.445033\pi$
$744$	0	0
$745$	34987.5	1.72060
$746$	0	0
$747$	4393.23	0.215181
$748$	0	0
$749$	7458.23	0.363842
$750$	0	0
$751$	15086.7	0.733051	0.366526	−	0.930408i	$-0.380547\pi$
$751$	15086.7	0.733051	0.366526	+	0.930408i	$0.380547\pi$
$752$	0	0
$753$	1065.79	0.0515800
$754$	0	0
$755$	−5893.61	−0.284093
$756$	0	0
$757$	9267.03	0.444935	0.222467	−	0.974940i	$-0.428589\pi$
$757$	9267.03	0.444935	0.222467	+	0.974940i	$0.428589\pi$
$758$	0	0
$759$	2147.85	0.102717
$760$	0	0
$761$	11490.5	0.547347	0.273674	−	0.961823i	$-0.411761\pi$
$761$	11490.5	0.547347	0.273674	+	0.961823i	$0.411761\pi$
$762$	0	0
$763$	15278.8	0.724939
$764$	0	0
$765$	−29277.8	−1.38371
$766$	0	0
$767$	5359.15	0.252291
$768$	0	0
$769$	2857.78	0.134011	0.0670054	−	0.997753i	$-0.478656\pi$
$769$	2857.78	0.134011	0.0670054	+	0.997753i	$0.478656\pi$
$770$	0	0
$771$	25210.2	1.17759
$772$	0	0
$773$	11841.4	0.550979	0.275490	−	0.961304i	$-0.411160\pi$
$773$	11841.4	0.550979	0.275490	+	0.961304i	$0.411160\pi$
$774$	0	0
$775$	54005.4	2.50314
$776$	0	0
$777$	23818.3	1.09971
$778$	0	0
$779$	5478.09	0.251955
$780$	0	0
$781$	−17130.5	−0.784862
$782$	0	0
$783$	1115.73	0.0509231
$784$	0	0
$785$	23243.2	1.05680
$786$	0	0
$787$	−3769.56	−0.170738	−0.0853688	−	0.996349i	$-0.527207\pi$
$787$	−3769.56	−0.170738	−0.0853688	+	0.996349i	$0.527207\pi$
$788$	0	0
$789$	−1743.29	−0.0786602
$790$	0	0
$791$	−11611.2	−0.521929
$792$	0	0
$793$	17455.5	0.781670
$794$	0	0
$795$	96744.7	4.31595
$796$	0	0
$797$	−34947.3	−1.55319	−0.776597	−	0.629997i	$-0.783056\pi$
$797$	−34947.3	−1.55319	−0.776597	+	0.629997i	$0.783056\pi$
$798$	0	0
$799$	18361.0	0.812973
$800$	0	0
$801$	−6552.39	−0.289035
$802$	0	0
$803$	2342.19	0.102932
$804$	0	0
$805$	3162.52	0.138465
$806$	0	0
$807$	4087.33	0.178291
$808$	0	0
$809$	37044.7	1.60992	0.804959	−	0.593331i	$-0.202187\pi$
$809$	37044.7	1.60992	0.804959	+	0.593331i	$0.202187\pi$
$810$	0	0
$811$	−24259.5	−1.05039	−0.525195	−	0.850982i	$-0.676008\pi$
$811$	−24259.5	−1.05039	−0.525195	+	0.850982i	$0.676008\pi$
$812$	0	0
$813$	−2787.65	−0.120255
$814$	0	0
$815$	−13473.5	−0.579088
$816$	0	0
$817$	26951.7	1.15413
$818$	0	0
$819$	−9054.58	−0.386316
$820$	0	0
$821$	28353.3	1.20528	0.602642	−	0.798012i	$-0.294115\pi$
$821$	28353.3	1.20528	0.602642	+	0.798012i	$0.294115\pi$
$822$	0	0
$823$	−1828.12	−0.0774291	−0.0387146	−	0.999250i	$-0.512326\pi$
$823$	−1828.12	−0.0774291	−0.0387146	+	0.999250i	$0.512326\pi$
$824$	0	0
$825$	29365.4	1.23924
$826$	0	0
$827$	40192.7	1.69001	0.845003	−	0.534761i	$-0.179598\pi$
$827$	40192.7	1.69001	0.845003	+	0.534761i	$0.179598\pi$
$828$	0	0
$829$	−1752.26	−0.0734120	−0.0367060	−	0.999326i	$-0.511687\pi$
$829$	−1752.26	−0.0734120	−0.0367060	+	0.999326i	$0.511687\pi$
$830$	0	0
$831$	51778.9	2.16148
$832$	0	0
$833$	−11574.2	−0.481418
$834$	0	0
$835$	−19011.3	−0.787920
$836$	0	0
$837$	9268.98	0.382775
$838$	0	0
$839$	−44190.0	−1.81837	−0.909183	−	0.416397i	$-0.863293\pi$
$839$	−44190.0	−1.81837	−0.909183	+	0.416397i	$0.863293\pi$
$840$	0	0
$841$	841.000	0.0344828
$842$	0	0
$843$	4122.86	0.168445
$844$	0	0
$845$	27020.0	1.10002
$846$	0	0
$847$	10737.3	0.435580
$848$	0	0
$849$	−15533.4	−0.627920
$850$	0	0
$851$	−4924.93	−0.198384
$852$	0	0
$853$	19032.1	0.763946	0.381973	−	0.924173i	$-0.375245\pi$
$853$	19032.1	0.763946	0.381973	+	0.924173i	$0.375245\pi$
$854$	0	0
$855$	31047.6	1.24188
$856$	0	0
$857$	15872.9	0.632682	0.316341	−	0.948646i	$-0.397546\pi$
$857$	15872.9	0.632682	0.316341	+	0.948646i	$0.397546\pi$
$858$	0	0
$859$	8861.23	0.351969	0.175984	−	0.984393i	$-0.443689\pi$
$859$	8861.23	0.351969	0.175984	+	0.984393i	$0.443689\pi$
$860$	0	0
$861$	8368.02	0.331221
$862$	0	0
$863$	−18131.8	−0.715195	−0.357598	−	0.933876i	$-0.616404\pi$
$863$	−18131.8	−0.715195	−0.357598	+	0.933876i	$0.616404\pi$
$864$	0	0
$865$	−22027.2	−0.865837
$866$	0	0
$867$	−19333.5	−0.757325
$868$	0	0
$869$	20438.0	0.797827
$870$	0	0
$871$	2242.13	0.0872236
$872$	0	0
$873$	15089.9	0.585012
$874$	0	0
$875$	19127.2	0.738992
$876$	0	0
$877$	3036.83	0.116929	0.0584644	−	0.998289i	$-0.481380\pi$
$877$	3036.83	0.116929	0.0584644	+	0.998289i	$0.481380\pi$
$878$	0	0
$879$	−63748.9	−2.44619
$880$	0	0
$881$	13059.2	0.499404	0.249702	−	0.968323i	$-0.419667\pi$
$881$	13059.2	0.499404	0.249702	+	0.968323i	$0.419667\pi$
$882$	0	0
$883$	12769.0	0.486650	0.243325	−	0.969945i	$-0.421762\pi$
$883$	12769.0	0.486650	0.243325	+	0.969945i	$0.421762\pi$
$884$	0	0
$885$	−28070.4	−1.06619
$886$	0	0
$887$	−4488.25	−0.169899	−0.0849496	−	0.996385i	$-0.527073\pi$
$887$	−4488.25	−0.169899	−0.0849496	+	0.996385i	$0.527073\pi$
$888$	0	0
$889$	24056.2	0.907557
$890$	0	0
$891$	−9698.13	−0.364646
$892$	0	0
$893$	−19470.9	−0.729642
$894$	0	0
$895$	36233.8	1.35325
$896$	0	0
$897$	3451.50	0.128475
$898$	0	0
$899$	6986.67	0.259198
$900$	0	0
$901$	−32992.4	−1.21991
$902$	0	0
$903$	41170.0	1.51722
$904$	0	0
$905$	−33214.6	−1.21999
$906$	0	0
$907$	−12432.5	−0.455144	−0.227572	−	0.973761i	$-0.573079\pi$
$907$	−12432.5	−0.455144	−0.227572	+	0.973761i	$0.573079\pi$
$908$	0	0
$909$	−53213.6	−1.94168
$910$	0	0
$911$	−28149.0	−1.02373	−0.511864	−	0.859066i	$-0.671045\pi$
$911$	−28149.0	−1.02373	−0.511864	+	0.859066i	$0.671045\pi$
$912$	0	0
$913$	2340.64	0.0848453
$914$	0	0
$915$	−91429.6	−3.30336
$916$	0	0
$917$	−10898.5	−0.392477
$918$	0	0
$919$	−22107.7	−0.793542	−0.396771	−	0.917918i	$-0.629869\pi$
$919$	−22107.7	−0.793542	−0.396771	+	0.917918i	$0.629869\pi$
$920$	0	0
$921$	56882.5	2.03512
$922$	0	0
$923$	−27527.9	−0.981682
$924$	0	0
$925$	−67333.7	−2.39342
$926$	0	0
$927$	−22493.3	−0.796954
$928$	0	0
$929$	35006.0	1.23629	0.618144	−	0.786065i	$-0.287885\pi$
$929$	35006.0	1.23629	0.618144	+	0.786065i	$0.287885\pi$
$930$	0	0
$931$	12273.8	0.432072
$932$	0	0
$933$	58152.2	2.04054
$934$	0	0
$935$	−15598.7	−0.545595
$936$	0	0
$937$	−1599.10	−0.0557529	−0.0278765	−	0.999611i	$-0.508875\pi$
$937$	−1599.10	−0.0557529	−0.0278765	+	0.999611i	$0.508875\pi$
$938$	0	0
$939$	63200.2	2.19644
$940$	0	0
$941$	−8849.46	−0.306572	−0.153286	−	0.988182i	$-0.548986\pi$
$941$	−8849.46	−0.306572	−0.153286	+	0.988182i	$0.548986\pi$
$942$	0	0
$943$	−1730.26	−0.0597510
$944$	0	0
$945$	7420.96	0.255454
$946$	0	0
$947$	19102.2	0.655480	0.327740	−	0.944768i	$-0.393713\pi$
$947$	19102.2	0.655480	0.327740	+	0.944768i	$0.393713\pi$
$948$	0	0
$949$	3763.80	0.128744
$950$	0	0
$951$	10721.8	0.365591
$952$	0	0
$953$	−37972.2	−1.29070	−0.645352	−	0.763886i	$-0.723289\pi$
$953$	−37972.2	−1.29070	−0.645352	+	0.763886i	$0.723289\pi$
$954$	0	0
$955$	62555.1	2.11962
$956$	0	0
$957$	3799.00	0.128322
$958$	0	0
$959$	25527.7	0.859575
$960$	0	0
$961$	28251.3	0.948318
$962$	0	0
$963$	−23126.7	−0.773882
$964$	0	0
$965$	76762.4	2.56069
$966$	0	0
$967$	−9884.51	−0.328712	−0.164356	−	0.986401i	$-0.552555\pi$
$967$	−9884.51	−0.328712	−0.164356	+	0.986401i	$0.552555\pi$
$968$	0	0
$969$	−19519.4	−0.647113
$970$	0	0
$971$	−44754.0	−1.47912	−0.739559	−	0.673091i	$-0.764966\pi$
$971$	−44754.0	−1.47912	−0.739559	+	0.673091i	$0.764966\pi$
$972$	0	0
$973$	10074.4	0.331933
$974$	0	0
$975$	47188.9	1.55000
$976$	0	0
$977$	−38370.4	−1.25648	−0.628238	−	0.778021i	$-0.716224\pi$
$977$	−38370.4	−1.25648	−0.628238	+	0.778021i	$0.716224\pi$
$978$	0	0
$979$	−3491.00	−0.113966
$980$	0	0
$981$	−47376.9	−1.54192
$982$	0	0
$983$	−682.565	−0.0221469	−0.0110735	−	0.999939i	$-0.503525\pi$
$983$	−682.565	−0.0221469	−0.0110735	+	0.999939i	$0.503525\pi$
$984$	0	0
$985$	−7117.50	−0.230236
$986$	0	0
$987$	−29742.7	−0.959190
$988$	0	0
$989$	−8512.76	−0.273701
$990$	0	0
$991$	−62176.4	−1.99304	−0.996518	−	0.0833725i	$-0.973431\pi$
$991$	−62176.4	−1.99304	−0.996518	+	0.0833725i	$0.973431\pi$
$992$	0	0
$993$	−10662.2	−0.340741
$994$	0	0
$995$	87179.0	2.77765
$996$	0	0
$997$	−18235.2	−0.579252	−0.289626	−	0.957140i	$-0.593531\pi$
$997$	−18235.2	−0.579252	−0.289626	+	0.957140i	$0.593531\pi$
$998$	0	0
$999$	−11556.5	−0.365998

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1856.4.a.q.1.3		3
4.3	odd	2		1856.4.a.t.1.1		3
8.3	odd	2		464.4.a.h.1.3		3
8.5	even	2		232.4.a.a.1.1	✓	3
24.5	odd	2		2088.4.a.a.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
232.4.a.a.1.1	✓	3	8.5	even	2
464.4.a.h.1.3		3	8.3	odd	2
1856.4.a.q.1.3		3	1.1	even	1	trivial
1856.4.a.t.1.1		3	4.3	odd	2
2088.4.a.a.1.1		3	24.5	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:	\( N \)	\(=\)	\( 1856 = 2^{6} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1856.a (trivial)

\(f(q)\)	\(=\)	\(q+7.68170 q^{3} -18.6859 q^{5} -10.3225 q^{7} +32.0084 q^{9} +O(q^{10})\)	\(q+7.68170 q^{3} -18.6859 q^{5} -10.3225 q^{7} +32.0084 q^{9} +17.0535 q^{11} +27.4043 q^{13} -143.540 q^{15} +48.9507 q^{17} -51.9098 q^{19} -79.2945 q^{21} +16.3958 q^{23} +224.163 q^{25} +38.4733 q^{27} +29.0000 q^{29} +240.920 q^{31} +131.000 q^{33} +192.886 q^{35} -300.378 q^{37} +210.511 q^{39} -105.531 q^{41} -519.203 q^{43} -598.107 q^{45} +375.092 q^{47} -236.445 q^{49} +376.024 q^{51} -673.993 q^{53} -318.661 q^{55} -398.755 q^{57} +195.559 q^{59} +636.965 q^{61} -330.408 q^{63} -512.074 q^{65} +81.8170 q^{67} +125.948 q^{69} -1004.51 q^{71} +137.343 q^{73} +1721.96 q^{75} -176.035 q^{77} +1198.46 q^{79} -568.688 q^{81} +137.252 q^{83} -914.689 q^{85} +222.769 q^{87} -204.708 q^{89} -282.881 q^{91} +1850.67 q^{93} +969.983 q^{95} +471.435 q^{97} +545.857 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{3} - 11 q^{5} - 38 q^{7} + 58 q^{9}+O(q^{10}) \) 3 * q - 3 * q^3 - 11 * q^5 - 38 * q^7 + 58 * q^9	\( 3 q - 3 q^{3} - 11 q^{5} - 38 q^{7} + 58 q^{9} - 65 q^{11} - 29 q^{13} - 85 q^{15} + 244 q^{17} - 312 q^{19} + 214 q^{21} - 24 q^{23} + 436 q^{25} - 57 q^{27} + 87 q^{29} + 249 q^{31} + 393 q^{33} + 718 q^{35} - 200 q^{37} + 749 q^{39} - 470 q^{41} - 97 q^{43} - 1562 q^{45} + 377 q^{47} + 383 q^{49} - 612 q^{51} - 1007 q^{53} - 1399 q^{55} + 1128 q^{57} + 364 q^{59} + 524 q^{61} - 2244 q^{63} + 301 q^{65} + 820 q^{67} + 824 q^{69} - 782 q^{71} + 1620 q^{73} + 1356 q^{75} - 158 q^{77} + 427 q^{79} + 227 q^{81} - 1520 q^{83} + 68 q^{85} - 87 q^{87} - 2474 q^{89} + 2002 q^{91} + 1807 q^{93} + 568 q^{95} - 170 q^{97} + 1362 q^{99}+O(q^{100}) \) 3 * q - 3 * q^3 - 11 * q^5 - 38 * q^7 + 58 * q^9 - 65 * q^11 - 29 * q^13 - 85 * q^15 + 244 * q^17 - 312 * q^19 + 214 * q^21 - 24 * q^23 + 436 * q^25 - 57 * q^27 + 87 * q^29 + 249 * q^31 + 393 * q^33 + 718 * q^35 - 200 * q^37 + 749 * q^39 - 470 * q^41 - 97 * q^43 - 1562 * q^45 + 377 * q^47 + 383 * q^49 - 612 * q^51 - 1007 * q^53 - 1399 * q^55 + 1128 * q^57 + 364 * q^59 + 524 * q^61 - 2244 * q^63 + 301 * q^65 + 820 * q^67 + 824 * q^69 - 782 * q^71 + 1620 * q^73 + 1356 * q^75 - 158 * q^77 + 427 * q^79 + 227 * q^81 - 1520 * q^83 + 68 * q^85 - 87 * q^87 - 2474 * q^89 + 2002 * q^91 + 1807 * q^93 + 568 * q^95 - 170 * q^97 + 1362 * q^99

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