LMFDB - Embedded newform 1856.4.a.bc.1.4

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:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 40x^{4} - 88x^{3} - 8x^{2} + 48x - 9 $ x^6 - 40x^4 - 88x^3 - 8x^2 + 48x - 9
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{7} $
Twist minimal:	no (minimal twist has level 232)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-4.70501$ 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)

$f(q)$	$=$	$q+0.586661 q^{3} -15.4201 q^{5} +28.5777 q^{7} -26.6558 q^{9} +O(q^{10})$	$q+0.586661 q^{3} -15.4201 q^{5} +28.5777 q^{7} -26.6558 q^{9} -63.9223 q^{11} -38.9302 q^{13} -9.04637 q^{15} -49.3308 q^{17} -95.2029 q^{19} +16.7654 q^{21} -208.417 q^{23} +112.779 q^{25} -31.4778 q^{27} +29.0000 q^{29} +182.420 q^{31} -37.5007 q^{33} -440.671 q^{35} +350.182 q^{37} -22.8388 q^{39} -147.936 q^{41} -1.65929 q^{43} +411.035 q^{45} -225.986 q^{47} +473.684 q^{49} -28.9405 q^{51} -460.380 q^{53} +985.688 q^{55} -55.8518 q^{57} -173.279 q^{59} +329.471 q^{61} -761.762 q^{63} +600.307 q^{65} -202.431 q^{67} -122.270 q^{69} +395.881 q^{71} +694.701 q^{73} +66.1632 q^{75} -1826.75 q^{77} +532.229 q^{79} +701.241 q^{81} +410.917 q^{83} +760.686 q^{85} +17.0132 q^{87} +105.742 q^{89} -1112.53 q^{91} +107.019 q^{93} +1468.04 q^{95} -993.097 q^{97} +1703.90 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 5 q^{3} + 5 q^{5} + 38 q^{7} + 47 q^{9}+O(q^{10}) $ 6 * q - 5 * q^3 + 5 * q^5 + 38 * q^7 + 47 * q^9	$ 6 q - 5 q^{3} + 5 q^{5} + 38 q^{7} + 47 q^{9} - 19 q^{11} - 13 q^{13} + 191 q^{15} - 218 q^{17} - 290 q^{19} + 266 q^{21} + 196 q^{23} - 13 q^{25} - 437 q^{27} + 174 q^{29} + 675 q^{31} + 291 q^{33} - 466 q^{35} + 238 q^{37} + 1297 q^{39} - 464 q^{41} - 579 q^{43} + 148 q^{45} + 975 q^{47} + 914 q^{49} - 576 q^{51} - 515 q^{53} + 1605 q^{55} - 340 q^{57} - 108 q^{59} - 1158 q^{61} + 1136 q^{63} + 1239 q^{65} - 80 q^{67} - 2568 q^{69} - 438 q^{71} + 262 q^{73} + 1766 q^{75} - 194 q^{77} + 237 q^{79} + 2554 q^{81} - 1288 q^{83} - 3112 q^{85} - 145 q^{87} - 252 q^{89} + 2450 q^{91} - 2131 q^{93} - 180 q^{95} + 380 q^{97} + 2264 q^{99}+O(q^{100}) $ 6 * q - 5 * q^3 + 5 * q^5 + 38 * q^7 + 47 * q^9 - 19 * q^11 - 13 * q^13 + 191 * q^15 - 218 * q^17 - 290 * q^19 + 266 * q^21 + 196 * q^23 - 13 * q^25 - 437 * q^27 + 174 * q^29 + 675 * q^31 + 291 * q^33 - 466 * q^35 + 238 * q^37 + 1297 * q^39 - 464 * q^41 - 579 * q^43 + 148 * q^45 + 975 * q^47 + 914 * q^49 - 576 * q^51 - 515 * q^53 + 1605 * q^55 - 340 * q^57 - 108 * q^59 - 1158 * q^61 + 1136 * q^63 + 1239 * q^65 - 80 * q^67 - 2568 * q^69 - 438 * q^71 + 262 * q^73 + 1766 * q^75 - 194 * q^77 + 237 * q^79 + 2554 * q^81 - 1288 * q^83 - 3112 * q^85 - 145 * q^87 - 252 * q^89 + 2450 * q^91 - 2131 * q^93 - 180 * q^95 + 380 * q^97 + 2264 * 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
$3$	0.586661	0.112903	0.0564515	−	0.998405i	$-0.482021\pi$
$3$	0.586661	0.112903	0.0564515	+	0.998405i	$0.482021\pi$
$4$	0	0
$5$	−15.4201	−1.37922	−0.689608	−	0.724183i	$-0.742217\pi$
$5$	−15.4201	−1.37922	−0.689608	+	0.724183i	$0.742217\pi$
$6$	0	0
$7$	28.5777	1.54305	0.771525	−	0.636199i	$-0.219494\pi$
$7$	28.5777	1.54305	0.771525	+	0.636199i	$0.219494\pi$
$8$	0	0
$9$	−26.6558	−0.987253
$10$	0	0
$11$	−63.9223	−1.75212	−0.876059	−	0.482204i	$-0.839837\pi$
$11$	−63.9223	−1.75212	−0.876059	+	0.482204i	$0.839837\pi$
$12$	0	0
$13$	−38.9302	−0.830561	−0.415281	−	0.909693i	$-0.636317\pi$
$13$	−38.9302	−0.830561	−0.415281	+	0.909693i	$0.636317\pi$
$14$	0	0
$15$	−9.04637	−0.155717
$16$	0	0
$17$	−49.3308	−0.703793	−0.351897	−	0.936039i	$-0.614463\pi$
$17$	−49.3308	−0.703793	−0.351897	+	0.936039i	$0.614463\pi$
$18$	0	0
$19$	−95.2029	−1.14953	−0.574764	−	0.818319i	$-0.694906\pi$
$19$	−95.2029	−1.14953	−0.574764	+	0.818319i	$0.694906\pi$
$20$	0	0
$21$	16.7654	0.174215
$22$	0	0
$23$	−208.417	−1.88947	−0.944737	−	0.327830i	$-0.893683\pi$
$23$	−208.417	−1.88947	−0.944737	+	0.327830i	$0.893683\pi$
$24$	0	0
$25$	112.779	0.902235
$26$	0	0
$27$	−31.4778	−0.224367
$28$	0	0
$29$	29.0000	0.185695
$29$	29.0000	0.185695
$30$	0	0
$31$	182.420	1.05689	0.528446	−	0.848967i	$-0.322775\pi$
$31$	182.420	1.05689	0.528446	+	0.848967i	$0.322775\pi$
$32$	0	0
$33$	−37.5007	−0.197819
$34$	0	0
$35$	−440.671	−2.12820
$36$	0	0
$37$	350.182	1.55593	0.777967	−	0.628305i	$-0.216251\pi$
$37$	350.182	1.55593	0.777967	+	0.628305i	$0.216251\pi$
$38$	0	0
$39$	−22.8388	−0.0937728
$40$	0	0
$41$	−147.936	−0.563506	−0.281753	−	0.959487i	$-0.590916\pi$
$41$	−147.936	−0.563506	−0.281753	+	0.959487i	$0.590916\pi$
$42$	0	0
$43$	−1.65929	−0.00588463	−0.00294232	−	0.999996i	$-0.500937\pi$
$43$	−1.65929	−0.00588463	−0.00294232	+	0.999996i	$0.500937\pi$
$44$	0	0
$45$	411.035	1.36163
$46$	0	0
$47$	−225.986	−0.701349	−0.350674	−	0.936497i	$-0.614048\pi$
$47$	−225.986	−0.701349	−0.350674	+	0.936497i	$0.614048\pi$
$48$	0	0
$49$	473.684	1.38100
$50$	0	0
$51$	−28.9405	−0.0794603
$52$	0	0
$53$	−460.380	−1.19317	−0.596586	−	0.802549i	$-0.703477\pi$
$53$	−460.380	−1.19317	−0.596586	+	0.802549i	$0.703477\pi$
$54$	0	0
$55$	985.688	2.41655
$56$	0	0
$57$	−55.8518	−0.129785
$58$	0	0
$59$	−173.279	−0.382356	−0.191178	−	0.981555i	$-0.561231\pi$
$59$	−173.279	−0.382356	−0.191178	+	0.981555i	$0.561231\pi$
$60$	0	0
$61$	329.471	0.691549	0.345774	−	0.938318i	$-0.387616\pi$
$61$	329.471	0.691549	0.345774	+	0.938318i	$0.387616\pi$
$62$	0	0
$63$	−761.762	−1.52338
$64$	0	0
$65$	600.307	1.14552
$66$	0	0
$67$	−202.431	−0.369117	−0.184559	−	0.982822i	$-0.559086\pi$
$67$	−202.431	−0.369117	−0.184559	+	0.982822i	$0.559086\pi$
$68$	0	0
$69$	−122.270	−0.213327
$70$	0	0
$71$	395.881	0.661724	0.330862	−	0.943679i	$-0.392660\pi$
$71$	395.881	0.661724	0.330862	+	0.943679i	$0.392660\pi$
$72$	0	0
$73$	694.701	1.11382	0.556909	−	0.830574i	$-0.311987\pi$
$73$	694.701	1.11382	0.556909	+	0.830574i	$0.311987\pi$
$74$	0	0
$75$	66.1632	0.101865
$76$	0	0
$77$	−1826.75	−2.70361
$78$	0	0
$79$	532.229	0.757981	0.378990	−	0.925401i	$-0.376271\pi$
$79$	532.229	0.757981	0.378990	+	0.925401i	$0.376271\pi$
$80$	0	0
$81$	701.241	0.961921
$82$	0	0
$83$	410.917	0.543421	0.271711	−	0.962379i	$-0.412411\pi$
$83$	410.917	0.543421	0.271711	+	0.962379i	$0.412411\pi$
$84$	0	0
$85$	760.686	0.970683
$86$	0	0
$87$	17.0132	0.0209655
$88$	0	0
$89$	105.742	0.125940	0.0629700	−	0.998015i	$-0.479943\pi$
$89$	105.742	0.125940	0.0629700	+	0.998015i	$0.479943\pi$
$90$	0	0
$91$	−1112.53	−1.28160
$92$	0	0
$93$	107.019	0.119326
$94$	0	0
$95$	1468.04	1.58545
$96$	0	0
$97$	−993.097	−1.03952	−0.519761	−	0.854312i	$-0.673979\pi$
$97$	−993.097	−1.03952	−0.519761	+	0.854312i	$0.673979\pi$
$98$	0	0
$99$	1703.90	1.72978
$100$	0	0
$101$	−1634.79	−1.61057	−0.805284	−	0.592890i	$-0.797987\pi$
$101$	−1634.79	−1.61057	−0.805284	+	0.592890i	$0.797987\pi$
$102$	0	0
$103$	−1382.18	−1.32223	−0.661116	−	0.750284i	$-0.729917\pi$
$103$	−1382.18	−1.32223	−0.661116	+	0.750284i	$0.729917\pi$
$104$	0	0
$105$	−258.524	−0.240280
$106$	0	0
$107$	−1648.59	−1.48949	−0.744746	−	0.667348i	$-0.767429\pi$
$107$	−1648.59	−1.48949	−0.744746	+	0.667348i	$0.767429\pi$
$108$	0	0
$109$	−4.74794	−0.00417220	−0.00208610	−	0.999998i	$-0.500664\pi$
$109$	−4.74794	−0.00417220	−0.00208610	+	0.999998i	$0.500664\pi$
$110$	0	0
$111$	205.438	0.175670
$112$	0	0
$113$	−2318.03	−1.92976	−0.964878	−	0.262699i	$-0.915387\pi$
$113$	−2318.03	−1.92976	−0.964878	+	0.262699i	$0.915387\pi$
$114$	0	0
$115$	3213.81	2.60599
$116$	0	0
$117$	1037.72	0.819974
$118$	0	0
$119$	−1409.76	−1.08599
$120$	0	0
$121$	2755.06	2.06992
$122$	0	0
$123$	−86.7884	−0.0636215
$124$	0	0
$125$	188.443	0.134839
$126$	0	0
$127$	−880.327	−0.615090	−0.307545	−	0.951534i	$-0.599507\pi$
$127$	−880.327	−0.615090	−0.307545	+	0.951534i	$0.599507\pi$
$128$	0	0
$129$	−0.973440	−0.000664392 0
$130$	0	0
$131$	2224.16	1.48340	0.741702	−	0.670730i	$-0.234019\pi$
$131$	2224.16	1.48340	0.741702	+	0.670730i	$0.234019\pi$
$132$	0	0
$133$	−2720.68	−1.77378
$134$	0	0
$135$	485.390	0.309450
$136$	0	0
$137$	721.323	0.449831	0.224915	−	0.974378i	$-0.427789\pi$
$137$	721.323	0.449831	0.224915	+	0.974378i	$0.427789\pi$
$138$	0	0
$139$	705.958	0.430781	0.215391	−	0.976528i	$-0.430898\pi$
$139$	705.958	0.430781	0.215391	+	0.976528i	$0.430898\pi$
$140$	0	0
$141$	−132.577	−0.0791843
$142$	0	0
$143$	2488.51	1.45524
$144$	0	0
$145$	−447.183	−0.256114
$146$	0	0
$147$	277.892	0.155919
$148$	0	0
$149$	1488.77	0.818557	0.409279	−	0.912409i	$-0.365780\pi$
$149$	1488.77	0.818557	0.409279	+	0.912409i	$0.365780\pi$
$150$	0	0
$151$	1561.95	0.841785	0.420892	−	0.907111i	$-0.361717\pi$
$151$	1561.95	0.841785	0.420892	+	0.907111i	$0.361717\pi$
$152$	0	0
$153$	1314.95	0.694822
$154$	0	0
$155$	−2812.94	−1.45768
$156$	0	0
$157$	−2511.51	−1.27669	−0.638346	−	0.769750i	$-0.720381\pi$
$157$	−2511.51	−1.27669	−0.638346	+	0.769750i	$0.720381\pi$
$158$	0	0
$159$	−270.087	−0.134713
$160$	0	0
$161$	−5956.07	−2.91555
$162$	0	0
$163$	711.605	0.341946	0.170973	−	0.985276i	$-0.445309\pi$
$163$	711.605	0.341946	0.170973	+	0.985276i	$0.445309\pi$
$164$	0	0
$165$	578.265	0.272835
$166$	0	0
$167$	3477.55	1.61138	0.805692	−	0.592335i	$-0.201794\pi$
$167$	3477.55	1.61138	0.805692	+	0.592335i	$0.201794\pi$
$168$	0	0
$169$	−681.440	−0.310168
$170$	0	0
$171$	2537.71	1.13488
$172$	0	0
$173$	2613.12	1.14839	0.574197	−	0.818717i	$-0.305314\pi$
$173$	2613.12	1.14839	0.574197	+	0.818717i	$0.305314\pi$
$174$	0	0
$175$	3222.97	1.39219
$176$	0	0
$177$	−101.656	−0.0431691
$178$	0	0
$179$	2932.10	1.22433	0.612166	−	0.790729i	$-0.290298\pi$
$179$	2932.10	1.22433	0.612166	+	0.790729i	$0.290298\pi$
$180$	0	0
$181$	−1235.51	−0.507376	−0.253688	−	0.967286i	$-0.581644\pi$
$181$	−1235.51	−0.507376	−0.253688	+	0.967286i	$0.581644\pi$
$182$	0	0
$183$	193.288	0.0780779
$184$	0	0
$185$	−5399.84	−2.14597
$186$	0	0
$187$	3153.34	1.23313
$188$	0	0
$189$	−899.562	−0.346209
$190$	0	0
$191$	−2281.95	−0.864484	−0.432242	−	0.901758i	$-0.642277\pi$
$191$	−2281.95	−0.864484	−0.432242	+	0.901758i	$0.642277\pi$
$192$	0	0
$193$	−1546.55	−0.576805	−0.288403	−	0.957509i	$-0.593124\pi$
$193$	−1546.55	−0.576805	−0.288403	+	0.957509i	$0.593124\pi$
$194$	0	0
$195$	352.177	0.129333
$196$	0	0
$197$	5026.48	1.81788	0.908939	−	0.416929i	$-0.136894\pi$
$197$	5026.48	1.81788	0.908939	+	0.416929i	$0.136894\pi$
$198$	0	0
$199$	−2621.88	−0.933972	−0.466986	−	0.884265i	$-0.654660\pi$
$199$	−2621.88	−0.933972	−0.466986	+	0.884265i	$0.654660\pi$
$200$	0	0
$201$	−118.758	−0.0416744
$202$	0	0
$203$	828.753	0.286537
$204$	0	0
$205$	2281.19	0.777197
$206$	0	0
$207$	5555.52	1.86539
$208$	0	0
$209$	6085.59	2.01411
$210$	0	0
$211$	−4872.90	−1.58988	−0.794939	−	0.606689i	$-0.792497\pi$
$211$	−4872.90	−1.58988	−0.794939	+	0.606689i	$0.792497\pi$
$212$	0	0
$213$	232.248	0.0747106
$214$	0	0
$215$	25.5864	0.00811618
$216$	0	0
$217$	5213.15	1.63084
$218$	0	0
$219$	407.554	0.125753
$220$	0	0
$221$	1920.46	0.584543
$222$	0	0
$223$	−3374.44	−1.01332	−0.506658	−	0.862147i	$-0.669119\pi$
$223$	−3374.44	−1.01332	−0.506658	+	0.862147i	$0.669119\pi$
$224$	0	0
$225$	−3006.23	−0.890734
$226$	0	0
$227$	1077.66	0.315097	0.157549	−	0.987511i	$-0.449641\pi$
$227$	1077.66	0.315097	0.157549	+	0.987511i	$0.449641\pi$
$228$	0	0
$229$	5976.56	1.72464	0.862319	−	0.506366i	$-0.169012\pi$
$229$	5976.56	1.72464	0.862319	+	0.506366i	$0.169012\pi$
$230$	0	0
$231$	−1071.68	−0.305245
$232$	0	0
$233$	3032.60	0.852671	0.426335	−	0.904565i	$-0.359804\pi$
$233$	3032.60	0.852671	0.426335	+	0.904565i	$0.359804\pi$
$234$	0	0
$235$	3484.72	0.967311
$236$	0	0
$237$	312.238	0.0855782
$238$	0	0
$239$	4444.48	1.20288	0.601442	−	0.798916i	$-0.294593\pi$
$239$	4444.48	1.20288	0.601442	+	0.798916i	$0.294593\pi$
$240$	0	0
$241$	−247.306	−0.0661010	−0.0330505	−	0.999454i	$-0.510522\pi$
$241$	−247.306	−0.0661010	−0.0330505	+	0.999454i	$0.510522\pi$
$242$	0	0
$243$	1261.29	0.332970
$244$	0	0
$245$	−7304.25	−1.90470
$246$	0	0
$247$	3706.27	0.954753
$248$	0	0
$249$	241.069	0.0613538
$250$	0	0
$251$	−465.961	−0.117176	−0.0585880	−	0.998282i	$-0.518660\pi$
$251$	−465.961	−0.117176	−0.0585880	+	0.998282i	$0.518660\pi$
$252$	0	0
$253$	13322.5	3.31058
$254$	0	0
$255$	446.265	0.109593
$256$	0	0
$257$	−1227.81	−0.298012	−0.149006	−	0.988836i	$-0.547607\pi$
$257$	−1227.81	−0.298012	−0.149006	+	0.988836i	$0.547607\pi$
$258$	0	0
$259$	10007.4	2.40088
$260$	0	0
$261$	−773.019	−0.183328
$262$	0	0
$263$	3503.69	0.821471	0.410735	−	0.911755i	$-0.365272\pi$
$263$	3503.69	0.821471	0.410735	+	0.911755i	$0.365272\pi$
$264$	0	0
$265$	7099.11	1.64564
$266$	0	0
$267$	62.0348	0.0142190
$268$	0	0
$269$	2548.93	0.577736	0.288868	−	0.957369i	$-0.406721\pi$
$269$	2548.93	0.577736	0.288868	+	0.957369i	$0.406721\pi$
$270$	0	0
$271$	2215.98	0.496720	0.248360	−	0.968668i	$-0.420108\pi$
$271$	2215.98	0.496720	0.248360	+	0.968668i	$0.420108\pi$
$272$	0	0
$273$	−652.681	−0.144696
$274$	0	0
$275$	−7209.12	−1.58082
$276$	0	0
$277$	−5622.98	−1.21968	−0.609842	−	0.792523i	$-0.708767\pi$
$277$	−5622.98	−1.21968	−0.609842	+	0.792523i	$0.708767\pi$
$278$	0	0
$279$	−4862.56	−1.04342
$280$	0	0
$281$	−1949.80	−0.413934	−0.206967	−	0.978348i	$-0.566359\pi$
$281$	−1949.80	−0.413934	−0.206967	+	0.978348i	$0.566359\pi$
$282$	0	0
$283$	−5741.24	−1.20594	−0.602970	−	0.797763i	$-0.706016\pi$
$283$	−5741.24	−1.20594	−0.602970	+	0.797763i	$0.706016\pi$
$284$	0	0
$285$	861.240	0.179002
$286$	0	0
$287$	−4227.68	−0.869519
$288$	0	0
$289$	−2479.47	−0.504675
$290$	0	0
$291$	−582.611	−0.117365
$292$	0	0
$293$	854.620	0.170401	0.0852004	−	0.996364i	$-0.472847\pi$
$293$	854.620	0.170401	0.0852004	+	0.996364i	$0.472847\pi$
$294$	0	0
$295$	2671.98	0.527351
$296$	0	0
$297$	2012.13	0.393117
$298$	0	0
$299$	8113.70	1.56932
$300$	0	0
$301$	−47.4186	−0.00908028
$302$	0	0
$303$	−959.065	−0.181838
$304$	0	0
$305$	−5080.48	−0.953795
$306$	0	0
$307$	2047.60	0.380661	0.190330	−	0.981720i	$-0.439044\pi$
$307$	2047.60	0.380661	0.190330	+	0.981720i	$0.439044\pi$
$308$	0	0
$309$	−810.868	−0.149284
$310$	0	0
$311$	−6046.79	−1.10251	−0.551257	−	0.834335i	$-0.685852\pi$
$311$	−6046.79	−1.10251	−0.551257	+	0.834335i	$0.685852\pi$
$312$	0	0
$313$	3748.54	0.676933	0.338467	−	0.940978i	$-0.390092\pi$
$313$	3748.54	0.676933	0.338467	+	0.940978i	$0.390092\pi$
$314$	0	0
$315$	11746.4	2.10107
$316$	0	0
$317$	−2048.42	−0.362936	−0.181468	−	0.983397i	$-0.558085\pi$
$317$	−2048.42	−0.362936	−0.181468	+	0.983397i	$0.558085\pi$
$318$	0	0
$319$	−1853.75	−0.325360
$320$	0	0
$321$	−967.166	−0.168168
$322$	0	0
$323$	4696.44	0.809030
$324$	0	0
$325$	−4390.52	−0.749361
$326$	0	0
$327$	−2.78543	−0.000471054 0
$328$	0	0
$329$	−6458.14	−1.08222
$330$	0	0
$331$	3977.02	0.660413	0.330207	−	0.943909i	$-0.392882\pi$
$331$	3977.02	0.660413	0.330207	+	0.943909i	$0.392882\pi$
$332$	0	0
$333$	−9334.39	−1.53610
$334$	0	0
$335$	3121.50	0.509092
$336$	0	0
$337$	−2881.00	−0.465691	−0.232846	−	0.972514i	$-0.574804\pi$
$337$	−2881.00	−0.465691	−0.232846	+	0.972514i	$0.574804\pi$
$338$	0	0
$339$	−1359.90	−0.217875
$340$	0	0
$341$	−11660.7	−1.85180
$342$	0	0
$343$	3734.65	0.587907
$344$	0	0
$345$	1885.41	0.294224
$346$	0	0
$347$	−5258.89	−0.813579	−0.406790	−	0.913522i	$-0.633352\pi$
$347$	−5258.89	−0.813579	−0.406790	+	0.913522i	$0.633352\pi$
$348$	0	0
$349$	−6025.28	−0.924143	−0.462071	−	0.886843i	$-0.652894\pi$
$349$	−6025.28	−0.924143	−0.462071	+	0.886843i	$0.652894\pi$
$350$	0	0
$351$	1225.44	0.186350
$352$	0	0
$353$	−10920.6	−1.64659	−0.823293	−	0.567617i	$-0.807865\pi$
$353$	−10920.6	−1.64659	−0.823293	+	0.567617i	$0.807865\pi$
$354$	0	0
$355$	−6104.52	−0.912660
$356$	0	0
$357$	−827.052	−0.122611
$358$	0	0
$359$	−142.924	−0.0210119	−0.0105059	−	0.999945i	$-0.503344\pi$
$359$	−142.924	−0.0210119	−0.0105059	+	0.999945i	$0.503344\pi$
$360$	0	0
$361$	2204.59	0.321415
$362$	0	0
$363$	1616.29	0.233700
$364$	0	0
$365$	−10712.4	−1.53619
$366$	0	0
$367$	4994.18	0.710338	0.355169	−	0.934802i	$-0.384423\pi$
$367$	4994.18	0.710338	0.355169	+	0.934802i	$0.384423\pi$
$368$	0	0
$369$	3943.36	0.556323
$370$	0	0
$371$	−13156.6	−1.84112
$372$	0	0
$373$	−5440.78	−0.755263	−0.377631	−	0.925956i	$-0.623261\pi$
$373$	−5440.78	−0.755263	−0.377631	+	0.925956i	$0.623261\pi$
$374$	0	0
$375$	110.552	0.0152237
$376$	0	0
$377$	−1128.98	−0.154231
$378$	0	0
$379$	1731.11	0.234620	0.117310	−	0.993095i	$-0.462573\pi$
$379$	1731.11	0.234620	0.117310	+	0.993095i	$0.462573\pi$
$380$	0	0
$381$	−516.454	−0.0694454
$382$	0	0
$383$	7323.08	0.977002	0.488501	−	0.872563i	$-0.337544\pi$
$383$	7323.08	0.977002	0.488501	+	0.872563i	$0.337544\pi$
$384$	0	0
$385$	28168.7	3.72886
$386$	0	0
$387$	44.2297	0.00580962
$388$	0	0
$389$	14764.8	1.92444	0.962219	−	0.272276i	$-0.0877765\pi$
$389$	14764.8	1.92444	0.962219	+	0.272276i	$0.0877765\pi$
$390$	0	0
$391$	10281.4	1.32980
$392$	0	0
$393$	1304.83	0.167481
$394$	0	0
$395$	−8207.03	−1.04542
$396$	0	0
$397$	3718.30	0.470066	0.235033	−	0.971987i	$-0.424480\pi$
$397$	3718.30	0.470066	0.235033	+	0.971987i	$0.424480\pi$
$398$	0	0
$399$	−1596.11	−0.200265
$400$	0	0
$401$	8586.13	1.06925	0.534627	−	0.845088i	$-0.320452\pi$
$401$	8586.13	1.06925	0.534627	+	0.845088i	$0.320452\pi$
$402$	0	0
$403$	−7101.66	−0.877813
$404$	0	0
$405$	−10813.2	−1.32670
$406$	0	0
$407$	−22384.4	−2.72618
$408$	0	0
$409$	−8326.22	−1.00661	−0.503307	−	0.864108i	$-0.667884\pi$
$409$	−8326.22	−1.00661	−0.503307	+	0.864108i	$0.667884\pi$
$410$	0	0
$411$	423.172	0.0507872
$412$	0	0
$413$	−4951.91	−0.589994
$414$	0	0
$415$	−6336.37	−0.749495
$416$	0	0
$417$	414.158	0.0486365
$418$	0	0
$419$	−768.872	−0.0896465	−0.0448232	−	0.998995i	$-0.514272\pi$
$419$	−768.872	−0.0896465	−0.0448232	+	0.998995i	$0.514272\pi$
$420$	0	0
$421$	−3147.83	−0.364408	−0.182204	−	0.983261i	$-0.558323\pi$
$421$	−3147.83	−0.364408	−0.182204	+	0.983261i	$0.558323\pi$
$422$	0	0
$423$	6023.83	0.692408
$424$	0	0
$425$	−5563.50	−0.634987
$426$	0	0
$427$	9415.53	1.06709
$428$	0	0
$429$	1459.91	0.164301
$430$	0	0
$431$	6418.90	0.717372	0.358686	−	0.933458i	$-0.383225\pi$
$431$	6418.90	0.717372	0.358686	+	0.933458i	$0.383225\pi$
$432$	0	0
$433$	−9273.92	−1.02928	−0.514638	−	0.857408i	$-0.672074\pi$
$433$	−9273.92	−1.02928	−0.514638	+	0.857408i	$0.672074\pi$
$434$	0	0
$435$	−262.345	−0.0289160
$436$	0	0
$437$	19841.9	2.17200
$438$	0	0
$439$	15038.5	1.63496	0.817481	−	0.575956i	$-0.195370\pi$
$439$	15038.5	1.63496	0.817481	+	0.575956i	$0.195370\pi$
$440$	0	0
$441$	−12626.4	−1.36340
$442$	0	0
$443$	6049.35	0.648788	0.324394	−	0.945922i	$-0.394840\pi$
$443$	6049.35	0.648788	0.324394	+	0.945922i	$0.394840\pi$
$444$	0	0
$445$	−1630.56	−0.173698
$446$	0	0
$447$	873.405	0.0924175
$448$	0	0
$449$	−15937.9	−1.67518	−0.837591	−	0.546297i	$-0.816037\pi$
$449$	−15937.9	−1.67518	−0.837591	+	0.546297i	$0.816037\pi$
$450$	0	0
$451$	9456.43	0.987330
$452$	0	0
$453$	916.333	0.0950399
$454$	0	0
$455$	17155.4	1.76760
$456$	0	0
$457$	15178.0	1.55361	0.776804	−	0.629742i	$-0.216840\pi$
$457$	15178.0	1.55361	0.776804	+	0.629742i	$0.216840\pi$
$458$	0	0
$459$	1552.83	0.157908
$460$	0	0
$461$	−4000.16	−0.404135	−0.202067	−	0.979372i	$-0.564766\pi$
$461$	−4000.16	−0.404135	−0.202067	+	0.979372i	$0.564766\pi$
$462$	0	0
$463$	9293.37	0.932829	0.466414	−	0.884566i	$-0.345546\pi$
$463$	9293.37	0.932829	0.466414	+	0.884566i	$0.345546\pi$
$464$	0	0
$465$	−1650.24	−0.164576
$466$	0	0
$467$	−1374.75	−0.136222	−0.0681111	−	0.997678i	$-0.521697\pi$
$467$	−1374.75	−0.136222	−0.0681111	+	0.997678i	$0.521697\pi$
$468$	0	0
$469$	−5785.00	−0.569566
$470$	0	0
$471$	−1473.41	−0.144142
$472$	0	0
$473$	106.066	0.0103106
$474$	0	0
$475$	−10736.9	−1.03714
$476$	0	0
$477$	12271.8	1.17796
$478$	0	0
$479$	4239.57	0.404406	0.202203	−	0.979344i	$-0.435190\pi$
$479$	4239.57	0.404406	0.202203	+	0.979344i	$0.435190\pi$
$480$	0	0
$481$	−13632.7	−1.29230
$482$	0	0
$483$	−3494.19	−0.329174
$484$	0	0
$485$	15313.6	1.43373
$486$	0	0
$487$	−11463.7	−1.06667	−0.533337	−	0.845903i	$-0.679062\pi$
$487$	−11463.7	−1.06667	−0.533337	+	0.845903i	$0.679062\pi$
$488$	0	0
$489$	417.471	0.0386067
$490$	0	0
$491$	3584.90	0.329500	0.164750	−	0.986335i	$-0.447318\pi$
$491$	3584.90	0.329500	0.164750	+	0.986335i	$0.447318\pi$
$492$	0	0
$493$	−1430.59	−0.130691
$494$	0	0
$495$	−26274.3	−2.38574
$496$	0	0
$497$	11313.4	1.02107
$498$	0	0
$499$	−2902.67	−0.260404	−0.130202	−	0.991487i	$-0.541563\pi$
$499$	−2902.67	−0.260404	−0.130202	+	0.991487i	$0.541563\pi$
$500$	0	0
$501$	2040.14	0.181930
$502$	0	0
$503$	1492.67	0.132316	0.0661578	−	0.997809i	$-0.478926\pi$
$503$	1492.67	0.132316	0.0661578	+	0.997809i	$0.478926\pi$
$504$	0	0
$505$	25208.6	2.22132
$506$	0	0
$507$	−399.774	−0.0350189
$508$	0	0
$509$	−7464.60	−0.650025	−0.325012	−	0.945710i	$-0.605368\pi$
$509$	−7464.60	−0.650025	−0.325012	+	0.945710i	$0.605368\pi$
$510$	0	0
$511$	19853.0	1.71868
$512$	0	0
$513$	2996.77	0.257916
$514$	0	0
$515$	21313.3	1.82364
$516$	0	0
$517$	14445.5	1.22885
$518$	0	0
$519$	1533.02	0.129657
$520$	0	0
$521$	−18000.8	−1.51368	−0.756841	−	0.653599i	$-0.773258\pi$
$521$	−18000.8	−1.51368	−0.756841	+	0.653599i	$0.773258\pi$
$522$	0	0
$523$	9092.63	0.760216	0.380108	−	0.924942i	$-0.375887\pi$
$523$	9092.63	0.760216	0.380108	+	0.924942i	$0.375887\pi$
$524$	0	0
$525$	1890.79	0.157183
$526$	0	0
$527$	−8998.95	−0.743834
$528$	0	0
$529$	31270.5	2.57011
$530$	0	0
$531$	4618.89	0.377482
$532$	0	0
$533$	5759.19	0.468027
$534$	0	0
$535$	25421.5	2.05433
$536$	0	0
$537$	1720.15	0.138231
$538$	0	0
$539$	−30279.0	−2.41968
$540$	0	0
$541$	−8128.88	−0.646003	−0.323002	−	0.946398i	$-0.604692\pi$
$541$	−8128.88	−0.646003	−0.323002	+	0.946398i	$0.604692\pi$
$542$	0	0
$543$	−724.828	−0.0572842
$544$	0	0
$545$	73.2137	0.00575437
$546$	0	0
$547$	−16181.0	−1.26481	−0.632405	−	0.774638i	$-0.717932\pi$
$547$	−16181.0	−1.26481	−0.632405	+	0.774638i	$0.717932\pi$
$548$	0	0
$549$	−8782.33	−0.682734
$550$	0	0
$551$	−2760.88	−0.213462
$552$	0	0
$553$	15209.9	1.16960
$554$	0	0
$555$	−3167.87	−0.242286
$556$	0	0
$557$	25415.5	1.93337	0.966687	−	0.255962i	$-0.0823923\pi$
$557$	25415.5	1.93337	0.966687	+	0.255962i	$0.0823923\pi$
$558$	0	0
$559$	64.5965	0.00488755
$560$	0	0
$561$	1849.94	0.139224
$562$	0	0
$563$	1126.26	0.0843096	0.0421548	−	0.999111i	$-0.486578\pi$
$563$	1126.26	0.0843096	0.0421548	+	0.999111i	$0.486578\pi$
$564$	0	0
$565$	35744.3	2.66155
$566$	0	0
$567$	20039.8	1.48429
$568$	0	0
$569$	4179.83	0.307957	0.153979	−	0.988074i	$-0.450791\pi$
$569$	4179.83	0.307957	0.153979	+	0.988074i	$0.450791\pi$
$570$	0	0
$571$	−14951.0	−1.09577	−0.547883	−	0.836555i	$-0.684566\pi$
$571$	−14951.0	−1.09577	−0.547883	+	0.836555i	$0.684566\pi$
$572$	0	0
$573$	−1338.73	−0.0976027
$574$	0	0
$575$	−23505.1	−1.70475
$576$	0	0
$577$	1745.68	0.125951	0.0629754	−	0.998015i	$-0.479941\pi$
$577$	1745.68	0.125951	0.0629754	+	0.998015i	$0.479941\pi$
$578$	0	0
$579$	−907.303	−0.0651230
$580$	0	0
$581$	11743.0	0.838526
$582$	0	0
$583$	29428.6	2.09058
$584$	0	0
$585$	−16001.7	−1.13092
$586$	0	0
$587$	9058.77	0.636960	0.318480	−	0.947930i	$-0.396828\pi$
$587$	9058.77	0.636960	0.318480	+	0.947930i	$0.396828\pi$
$588$	0	0
$589$	−17366.9	−1.21493
$590$	0	0
$591$	2948.84	0.205244
$592$	0	0
$593$	−15996.7	−1.10777	−0.553885	−	0.832593i	$-0.686855\pi$
$593$	−15996.7	−1.10777	−0.553885	+	0.832593i	$0.686855\pi$
$594$	0	0
$595$	21738.7	1.49781
$596$	0	0
$597$	−1538.16	−0.105448
$598$	0	0
$599$	−20371.2	−1.38955	−0.694777	−	0.719225i	$-0.744497\pi$
$599$	−20371.2	−1.38955	−0.694777	+	0.719225i	$0.744497\pi$
$600$	0	0
$601$	−13338.1	−0.905277	−0.452639	−	0.891694i	$-0.649517\pi$
$601$	−13338.1	−0.905277	−0.452639	+	0.891694i	$0.649517\pi$
$602$	0	0
$603$	5395.96	0.364412
$604$	0	0
$605$	−42483.3	−2.85486
$606$	0	0
$607$	−23051.7	−1.54142	−0.770709	−	0.637187i	$-0.780098\pi$
$607$	−23051.7	−1.54142	−0.770709	+	0.637187i	$0.780098\pi$
$608$	0	0
$609$	486.197	0.0323509
$610$	0	0
$611$	8797.66	0.582513
$612$	0	0
$613$	3532.33	0.232740	0.116370	−	0.993206i	$-0.462874\pi$
$613$	3532.33	0.232740	0.116370	+	0.993206i	$0.462874\pi$
$614$	0	0
$615$	1338.29	0.0877478
$616$	0	0
$617$	4491.89	0.293090	0.146545	−	0.989204i	$-0.453185\pi$
$617$	4491.89	0.293090	0.146545	+	0.989204i	$0.453185\pi$
$618$	0	0
$619$	23772.1	1.54359	0.771794	−	0.635872i	$-0.219359\pi$
$619$	23772.1	1.54359	0.771794	+	0.635872i	$0.219359\pi$
$620$	0	0
$621$	6560.49	0.423935
$622$	0	0
$623$	3021.87	0.194332
$624$	0	0
$625$	−17003.2	−1.08821
$626$	0	0
$627$	3570.18	0.227399
$628$	0	0
$629$	−17274.8	−1.09506
$630$	0	0
$631$	23199.8	1.46366	0.731830	−	0.681487i	$-0.238667\pi$
$631$	23199.8	1.46366	0.731830	+	0.681487i	$0.238667\pi$
$632$	0	0
$633$	−2858.74	−0.179502
$634$	0	0
$635$	13574.7	0.848341
$636$	0	0
$637$	−18440.6	−1.14701
$638$	0	0
$639$	−10552.5	−0.653289
$640$	0	0
$641$	−7281.96	−0.448705	−0.224353	−	0.974508i	$-0.572027\pi$
$641$	−7281.96	−0.448705	−0.224353	+	0.974508i	$0.572027\pi$
$642$	0	0
$643$	7917.29	0.485579	0.242790	−	0.970079i	$-0.421938\pi$
$643$	7917.29	0.485579	0.242790	+	0.970079i	$0.421938\pi$
$644$	0	0
$645$	15.0105	0.000916340 0
$646$	0	0
$647$	−21033.6	−1.27808	−0.639039	−	0.769174i	$-0.720668\pi$
$647$	−21033.6	−1.27808	−0.639039	+	0.769174i	$0.720668\pi$
$648$	0	0
$649$	11076.4	0.669932
$650$	0	0
$651$	3058.35	0.184126
$652$	0	0
$653$	7536.16	0.451628	0.225814	−	0.974170i	$-0.427496\pi$
$653$	7536.16	0.451628	0.225814	+	0.974170i	$0.427496\pi$
$654$	0	0
$655$	−34296.8	−2.04593
$656$	0	0
$657$	−18517.8	−1.09962
$658$	0	0
$659$	6740.56	0.398444	0.199222	−	0.979954i	$-0.436158\pi$
$659$	6740.56	0.398444	0.199222	+	0.979954i	$0.436158\pi$
$660$	0	0
$661$	−19676.2	−1.15782	−0.578908	−	0.815393i	$-0.696521\pi$
$661$	−19676.2	−1.15782	−0.578908	+	0.815393i	$0.696521\pi$
$662$	0	0
$663$	1126.66	0.0659967
$664$	0	0
$665$	41953.1	2.44642
$666$	0	0
$667$	−6044.08	−0.350866
$668$	0	0
$669$	−1979.65	−0.114406
$670$	0	0
$671$	−21060.6	−1.21168
$672$	0	0
$673$	−22947.0	−1.31433	−0.657164	−	0.753748i	$-0.728244\pi$
$673$	−22947.0	−1.31433	−0.657164	+	0.753748i	$0.728244\pi$
$674$	0	0
$675$	−3550.04	−0.202431
$676$	0	0
$677$	30162.1	1.71229	0.856146	−	0.516734i	$-0.172852\pi$
$677$	30162.1	1.71229	0.856146	+	0.516734i	$0.172852\pi$
$678$	0	0
$679$	−28380.4	−1.60404
$680$	0	0
$681$	632.223	0.0355754
$682$	0	0
$683$	−1124.17	−0.0629799	−0.0314900	−	0.999504i	$-0.510025\pi$
$683$	−1124.17	−0.0629799	−0.0314900	+	0.999504i	$0.510025\pi$
$684$	0	0
$685$	−11122.9	−0.620413
$686$	0	0
$687$	3506.21	0.194717
$688$	0	0
$689$	17922.7	0.991002
$690$	0	0
$691$	8429.01	0.464044	0.232022	−	0.972711i	$-0.425466\pi$
$691$	8429.01	0.464044	0.232022	+	0.972711i	$0.425466\pi$
$692$	0	0
$693$	48693.6	2.66914
$694$	0	0
$695$	−10885.9	−0.594140
$696$	0	0
$697$	7297.82	0.396592
$698$	0	0
$699$	1779.11	0.0962690
$700$	0	0
$701$	11175.3	0.602120	0.301060	−	0.953605i	$-0.402660\pi$
$701$	11175.3	0.602120	0.301060	+	0.953605i	$0.402660\pi$
$702$	0	0
$703$	−33338.3	−1.78859
$704$	0	0
$705$	2044.35	0.109212
$706$	0	0
$707$	−46718.4	−2.48519
$708$	0	0
$709$	−26940.5	−1.42704	−0.713519	−	0.700636i	$-0.752900\pi$
$709$	−26940.5	−1.42704	−0.713519	+	0.700636i	$0.752900\pi$
$710$	0	0
$711$	−14187.0	−0.748319
$712$	0	0
$713$	−38019.4	−1.99697
$714$	0	0
$715$	−38373.0	−2.00709
$716$	0	0
$717$	2607.40	0.135809
$718$	0	0
$719$	25489.4	1.32211	0.661054	−	0.750338i	$-0.270109\pi$
$719$	25489.4	1.32211	0.661054	+	0.750338i	$0.270109\pi$
$720$	0	0
$721$	−39499.4	−2.04027
$722$	0	0
$723$	−145.084	−0.00746300
$724$	0	0
$725$	3270.60	0.167541
$726$	0	0
$727$	−4219.71	−0.215269	−0.107634	−	0.994191i	$-0.534328\pi$
$727$	−4219.71	−0.215269	−0.107634	+	0.994191i	$0.534328\pi$
$728$	0	0
$729$	−18193.5	−0.924328
$730$	0	0
$731$	81.8541	0.00414157
$732$	0	0
$733$	−22.2883	−0.00112311	−0.000561554	−	1.00000i	$-0.500179\pi$
$733$	−22.2883	−0.00112311	−0.000561554		1.00000i	$0.500179\pi$
$734$	0	0
$735$	−4285.12	−0.215046
$736$	0	0
$737$	12939.8	0.646737
$738$	0	0
$739$	13298.7	0.661978	0.330989	−	0.943635i	$-0.392618\pi$
$739$	13298.7	0.661978	0.330989	+	0.943635i	$0.392618\pi$
$740$	0	0
$741$	2174.32	0.107794
$742$	0	0
$743$	−26379.6	−1.30252	−0.651261	−	0.758854i	$-0.725760\pi$
$743$	−26379.6	−1.30252	−0.651261	+	0.758854i	$0.725760\pi$
$744$	0	0
$745$	−22957.0	−1.12897
$746$	0	0
$747$	−10953.3	−0.536494
$748$	0	0
$749$	−47113.0	−2.29836
$750$	0	0
$751$	4123.41	0.200353	0.100177	−	0.994970i	$-0.468059\pi$
$751$	4123.41	0.200353	0.100177	+	0.994970i	$0.468059\pi$
$752$	0	0
$753$	−273.361	−0.0132295
$754$	0	0
$755$	−24085.4	−1.16100
$756$	0	0
$757$	−9980.93	−0.479211	−0.239606	−	0.970870i	$-0.577018\pi$
$757$	−9980.93	−0.479211	−0.239606	+	0.970870i	$0.577018\pi$
$758$	0	0
$759$	7815.78	0.373774
$760$	0	0
$761$	23211.6	1.10568	0.552838	−	0.833289i	$-0.313545\pi$
$761$	23211.6	1.10568	0.552838	+	0.833289i	$0.313545\pi$
$762$	0	0
$763$	−135.685	−0.00643792
$764$	0	0
$765$	−20276.7	−0.958309
$766$	0	0
$767$	6745.78	0.317570
$768$	0	0
$769$	11978.6	0.561716	0.280858	−	0.959749i	$-0.409381\pi$
$769$	11978.6	0.561716	0.280858	+	0.959749i	$0.409381\pi$
$770$	0	0
$771$	−720.311	−0.0336464
$772$	0	0
$773$	40266.3	1.87358	0.936790	−	0.349891i	$-0.113781\pi$
$773$	40266.3	1.87358	0.936790	+	0.349891i	$0.113781\pi$
$774$	0	0
$775$	20573.2	0.953565
$776$	0	0
$777$	5870.94	0.271067
$778$	0	0
$779$	14084.0	0.647767
$780$	0	0
$781$	−25305.6	−1.15942
$782$	0	0
$783$	−912.855	−0.0416638
$784$	0	0
$785$	38727.8	1.76083
$786$	0	0
$787$	22105.0	1.00122	0.500610	−	0.865673i	$-0.333109\pi$
$787$	22105.0	1.00122	0.500610	+	0.865673i	$0.333109\pi$
$788$	0	0
$789$	2055.48	0.0927464
$790$	0	0
$791$	−66244.1	−2.97771
$792$	0	0
$793$	−12826.4	−0.574374
$794$	0	0
$795$	4164.77	0.185798
$796$	0	0
$797$	−1056.63	−0.0469606	−0.0234803	−	0.999724i	$-0.507475\pi$
$797$	−1056.63	−0.0469606	−0.0234803	+	0.999724i	$0.507475\pi$
$798$	0	0
$799$	11148.1	0.493604
$800$	0	0
$801$	−2818.65	−0.124335
$802$	0	0
$803$	−44406.9	−1.95154
$804$	0	0
$805$	91843.1	4.02117
$806$	0	0
$807$	1495.36	0.0652281
$808$	0	0
$809$	26652.4	1.15828	0.579139	−	0.815229i	$-0.303389\pi$
$809$	26652.4	1.15828	0.579139	+	0.815229i	$0.303389\pi$
$810$	0	0
$811$	−25367.0	−1.09834	−0.549171	−	0.835710i	$-0.685057\pi$
$811$	−25367.0	−1.09834	−0.549171	+	0.835710i	$0.685057\pi$
$812$	0	0
$813$	1300.03	0.0560811
$814$	0	0
$815$	−10973.0	−0.471617
$816$	0	0
$817$	157.969	0.00676455
$818$	0	0
$819$	29655.5	1.26526
$820$	0	0
$821$	−36324.3	−1.54413	−0.772063	−	0.635546i	$-0.780775\pi$
$821$	−36324.3	−1.54413	−0.772063	+	0.635546i	$0.780775\pi$
$822$	0	0
$823$	−27173.2	−1.15091	−0.575456	−	0.817833i	$-0.695175\pi$
$823$	−27173.2	−1.15091	−0.575456	+	0.817833i	$0.695175\pi$
$824$	0	0
$825$	−4229.31	−0.178479
$826$	0	0
$827$	−33926.0	−1.42651	−0.713254	−	0.700906i	$-0.752779\pi$
$827$	−33926.0	−1.42651	−0.713254	+	0.700906i	$0.752779\pi$
$828$	0	0
$829$	−46809.1	−1.96110	−0.980548	−	0.196281i	$-0.937113\pi$
$829$	−46809.1	−1.96110	−0.980548	+	0.196281i	$0.937113\pi$
$830$	0	0
$831$	−3298.78	−0.137706
$832$	0	0
$833$	−23367.2	−0.971941
$834$	0	0
$835$	−53624.2	−2.22245
$836$	0	0
$837$	−5742.18	−0.237131
$838$	0	0
$839$	1470.77	0.0605202	0.0302601	−	0.999542i	$-0.490366\pi$
$839$	1470.77	0.0605202	0.0302601	+	0.999542i	$0.490366\pi$
$840$	0	0
$841$	841.000	0.0344828
$842$	0	0
$843$	−1143.87	−0.0467343
$844$	0	0
$845$	10507.9	0.427789
$846$	0	0
$847$	78733.3	3.19399
$848$	0	0
$849$	−3368.16	−0.136154
$850$	0	0
$851$	−72983.8	−2.93990
$852$	0	0
$853$	−30173.6	−1.21116	−0.605582	−	0.795783i	$-0.707060\pi$
$853$	−30173.6	−1.21116	−0.605582	+	0.795783i	$0.707060\pi$
$854$	0	0
$855$	−39131.8	−1.56524
$856$	0	0
$857$	35819.5	1.42774	0.713868	−	0.700280i	$-0.246942\pi$
$857$	35819.5	1.42774	0.713868	+	0.700280i	$0.246942\pi$
$858$	0	0
$859$	21551.9	0.856043	0.428021	−	0.903769i	$-0.359211\pi$
$859$	21551.9	0.856043	0.428021	+	0.903769i	$0.359211\pi$
$860$	0	0
$861$	−2480.21	−0.0981712
$862$	0	0
$863$	−948.634	−0.0374182	−0.0187091	−	0.999825i	$-0.505956\pi$
$863$	−948.634	−0.0374182	−0.0187091	+	0.999825i	$0.505956\pi$
$864$	0	0
$865$	−40294.6	−1.58388
$866$	0	0
$867$	−1454.61	−0.0569793
$868$	0	0
$869$	−34021.3	−1.32807
$870$	0	0
$871$	7880.67	0.306574
$872$	0	0
$873$	26471.8	1.02627
$874$	0	0
$875$	5385.28	0.208064
$876$	0	0
$877$	6462.02	0.248811	0.124405	−	0.992231i	$-0.460298\pi$
$877$	6462.02	0.248811	0.124405	+	0.992231i	$0.460298\pi$
$878$	0	0
$879$	501.372	0.0192387
$880$	0	0
$881$	−8788.14	−0.336072	−0.168036	−	0.985781i	$-0.553743\pi$
$881$	−8788.14	−0.336072	−0.168036	+	0.985781i	$0.553743\pi$
$882$	0	0
$883$	4250.34	0.161988	0.0809940	−	0.996715i	$-0.474191\pi$
$883$	4250.34	0.161988	0.0809940	+	0.996715i	$0.474191\pi$
$884$	0	0
$885$	1567.54	0.0595394
$886$	0	0
$887$	−29453.0	−1.11492	−0.557461	−	0.830203i	$-0.688224\pi$
$887$	−29453.0	−1.11492	−0.557461	+	0.830203i	$0.688224\pi$
$888$	0	0
$889$	−25157.7	−0.949114
$890$	0	0
$891$	−44824.9	−1.68540
$892$	0	0
$893$	21514.5	0.806220
$894$	0	0
$895$	−45213.3	−1.68862
$896$	0	0
$897$	4759.99	0.177181
$898$	0	0
$899$	5290.19	0.196260
$900$	0	0
$901$	22711.0	0.839747
$902$	0	0
$903$	−27.8187	−0.00102519
$904$	0	0
$905$	19051.7	0.699781
$906$	0	0
$907$	−40466.4	−1.48144	−0.740719	−	0.671814i	$-0.765515\pi$
$907$	−40466.4	−1.48144	−0.740719	+	0.671814i	$0.765515\pi$
$908$	0	0
$909$	43576.6	1.59004
$910$	0	0
$911$	29495.3	1.07269	0.536347	−	0.843998i	$-0.319804\pi$
$911$	29495.3	1.07269	0.536347	+	0.843998i	$0.319804\pi$
$912$	0	0
$913$	−26266.7	−0.952138
$914$	0	0
$915$	−2980.52	−0.107686
$916$	0	0
$917$	63561.4	2.28897
$918$	0	0
$919$	10000.9	0.358975	0.179487	−	0.983760i	$-0.442556\pi$
$919$	10000.9	0.358975	0.179487	+	0.983760i	$0.442556\pi$
$920$	0	0
$921$	1201.25	0.0429777
$922$	0	0
$923$	−15411.7	−0.549602
$924$	0	0
$925$	39493.3	1.40382
$926$	0	0
$927$	36843.0	1.30538
$928$	0	0
$929$	−17951.8	−0.633992	−0.316996	−	0.948427i	$-0.602674\pi$
$929$	−17951.8	−0.633992	−0.316996	+	0.948427i	$0.602674\pi$
$930$	0	0
$931$	−45096.1	−1.58750
$932$	0	0
$933$	−3547.41	−0.124477
$934$	0	0
$935$	−48624.8	−1.70075
$936$	0	0
$937$	27933.3	0.973895	0.486948	−	0.873431i	$-0.338110\pi$
$937$	27933.3	0.973895	0.486948	+	0.873431i	$0.338110\pi$
$938$	0	0
$939$	2199.12	0.0764278
$940$	0	0
$941$	−51039.5	−1.76816	−0.884080	−	0.467335i	$-0.845214\pi$
$941$	−51039.5	−1.76816	−0.884080	+	0.467335i	$0.845214\pi$
$942$	0	0
$943$	30832.4	1.06473
$944$	0	0
$945$	13871.3	0.477497
$946$	0	0
$947$	37607.4	1.29047	0.645236	−	0.763983i	$-0.276759\pi$
$947$	37607.4	1.29047	0.645236	+	0.763983i	$0.276759\pi$
$948$	0	0
$949$	−27044.9	−0.925093
$950$	0	0
$951$	−1201.73	−0.0409765
$952$	0	0
$953$	−31816.0	−1.08145	−0.540725	−	0.841199i	$-0.681850\pi$
$953$	−31816.0	−1.08145	−0.540725	+	0.841199i	$0.681850\pi$
$954$	0	0
$955$	35187.9	1.19231
$956$	0	0
$957$	−1087.52	−0.0367341
$958$	0	0
$959$	20613.7	0.694111
$960$	0	0
$961$	3486.16	0.117020
$962$	0	0
$963$	43944.7	1.47051
$964$	0	0
$965$	23848.0	0.795539
$966$	0	0
$967$	40630.4	1.35117	0.675587	−	0.737280i	$-0.263890\pi$
$967$	40630.4	1.35117	0.675587	+	0.737280i	$0.263890\pi$
$968$	0	0
$969$	2755.22	0.0913419
$970$	0	0
$971$	24235.4	0.800978	0.400489	−	0.916302i	$-0.368840\pi$
$971$	24235.4	0.800978	0.400489	+	0.916302i	$0.368840\pi$
$972$	0	0
$973$	20174.7	0.664717
$974$	0	0
$975$	−2575.75	−0.0846051
$976$	0	0
$977$	−6553.18	−0.214591	−0.107295	−	0.994227i	$-0.534219\pi$
$977$	−6553.18	−0.214591	−0.107295	+	0.994227i	$0.534219\pi$
$978$	0	0
$979$	−6759.29	−0.220662
$980$	0	0
$981$	126.560	0.00411902
$982$	0	0
$983$	−15778.3	−0.511952	−0.255976	−	0.966683i	$-0.582397\pi$
$983$	−15778.3	−0.511952	−0.255976	+	0.966683i	$0.582397\pi$
$984$	0	0
$985$	−77508.8	−2.50724
$986$	0	0
$987$	−3788.74	−0.122185
$988$	0	0
$989$	345.824	0.0111189
$990$	0	0
$991$	−41724.8	−1.33747	−0.668734	−	0.743502i	$-0.733163\pi$
$991$	−41724.8	−1.33747	−0.668734	+	0.743502i	$0.733163\pi$
$992$	0	0
$993$	2333.16	0.0745626
$994$	0	0
$995$	40429.7	1.28815
$996$	0	0
$997$	−5932.02	−0.188434	−0.0942171	−	0.995552i	$-0.530035\pi$
$997$	−5932.02	−0.188434	−0.0942171	+	0.995552i	$0.530035\pi$
$998$	0	0
$999$	−11022.9	−0.349100

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1856.4.a.bc.1.4		6
4.3	odd	2		1856.4.a.bd.1.3		6
8.3	odd	2		232.4.a.e.1.4	✓	6
8.5	even	2		464.4.a.n.1.3		6
24.11	even	2		2088.4.a.l.1.1		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
232.4.a.e.1.4	✓	6	8.3	odd	2
464.4.a.n.1.3		6	8.5	even	2
1856.4.a.bc.1.4		6	1.1	even	1	trivial
1856.4.a.bd.1.3		6	4.3	odd	2
2088.4.a.l.1.1		6	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)

\(f(q)\)	\(=\)	\(q+0.586661 q^{3} -15.4201 q^{5} +28.5777 q^{7} -26.6558 q^{9} +O(q^{10})\)	\(q+0.586661 q^{3} -15.4201 q^{5} +28.5777 q^{7} -26.6558 q^{9} -63.9223 q^{11} -38.9302 q^{13} -9.04637 q^{15} -49.3308 q^{17} -95.2029 q^{19} +16.7654 q^{21} -208.417 q^{23} +112.779 q^{25} -31.4778 q^{27} +29.0000 q^{29} +182.420 q^{31} -37.5007 q^{33} -440.671 q^{35} +350.182 q^{37} -22.8388 q^{39} -147.936 q^{41} -1.65929 q^{43} +411.035 q^{45} -225.986 q^{47} +473.684 q^{49} -28.9405 q^{51} -460.380 q^{53} +985.688 q^{55} -55.8518 q^{57} -173.279 q^{59} +329.471 q^{61} -761.762 q^{63} +600.307 q^{65} -202.431 q^{67} -122.270 q^{69} +395.881 q^{71} +694.701 q^{73} +66.1632 q^{75} -1826.75 q^{77} +532.229 q^{79} +701.241 q^{81} +410.917 q^{83} +760.686 q^{85} +17.0132 q^{87} +105.742 q^{89} -1112.53 q^{91} +107.019 q^{93} +1468.04 q^{95} -993.097 q^{97} +1703.90 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 5 q^{3} + 5 q^{5} + 38 q^{7} + 47 q^{9}+O(q^{10}) \) 6 * q - 5 * q^3 + 5 * q^5 + 38 * q^7 + 47 * q^9	\( 6 q - 5 q^{3} + 5 q^{5} + 38 q^{7} + 47 q^{9} - 19 q^{11} - 13 q^{13} + 191 q^{15} - 218 q^{17} - 290 q^{19} + 266 q^{21} + 196 q^{23} - 13 q^{25} - 437 q^{27} + 174 q^{29} + 675 q^{31} + 291 q^{33} - 466 q^{35} + 238 q^{37} + 1297 q^{39} - 464 q^{41} - 579 q^{43} + 148 q^{45} + 975 q^{47} + 914 q^{49} - 576 q^{51} - 515 q^{53} + 1605 q^{55} - 340 q^{57} - 108 q^{59} - 1158 q^{61} + 1136 q^{63} + 1239 q^{65} - 80 q^{67} - 2568 q^{69} - 438 q^{71} + 262 q^{73} + 1766 q^{75} - 194 q^{77} + 237 q^{79} + 2554 q^{81} - 1288 q^{83} - 3112 q^{85} - 145 q^{87} - 252 q^{89} + 2450 q^{91} - 2131 q^{93} - 180 q^{95} + 380 q^{97} + 2264 q^{99}+O(q^{100}) \) 6 * q - 5 * q^3 + 5 * q^5 + 38 * q^7 + 47 * q^9 - 19 * q^11 - 13 * q^13 + 191 * q^15 - 218 * q^17 - 290 * q^19 + 266 * q^21 + 196 * q^23 - 13 * q^25 - 437 * q^27 + 174 * q^29 + 675 * q^31 + 291 * q^33 - 466 * q^35 + 238 * q^37 + 1297 * q^39 - 464 * q^41 - 579 * q^43 + 148 * q^45 + 975 * q^47 + 914 * q^49 - 576 * q^51 - 515 * q^53 + 1605 * q^55 - 340 * q^57 - 108 * q^59 - 1158 * q^61 + 1136 * q^63 + 1239 * q^65 - 80 * q^67 - 2568 * q^69 - 438 * q^71 + 262 * q^73 + 1766 * q^75 - 194 * q^77 + 237 * q^79 + 2554 * q^81 - 1288 * q^83 - 3112 * q^85 - 145 * q^87 - 252 * q^89 + 2450 * q^91 - 2131 * q^93 - 180 * q^95 + 380 * q^97 + 2264 * q^99

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