LMFDB - Embedded newform 1856.4.a.bg.1.6

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:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - 4x^{8} - 138x^{7} + 394x^{6} + 5872x^{5} - 10822x^{4} - 85158x^{3} + 30654x^{2} + 439999x + 396802 $ x^9 - 4x^8 - 138x^7 + 394x^6 + 5872x^5 - 10822x^4 - 85158x^3 + 30654x^2 + 439999x + 396802
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{12} $
Twist minimal:	no (minimal twist has level 928)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$3.39022$ of defining polynomial
Character	$\chi$	$=$	1856.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.39022 q^{3} +19.3767 q^{5} +7.92881 q^{7} -15.5064 q^{9} +O(q^{10})$	$q+3.39022 q^{3} +19.3767 q^{5} +7.92881 q^{7} -15.5064 q^{9} -15.4230 q^{11} -45.5894 q^{13} +65.6911 q^{15} -60.6323 q^{17} -79.4211 q^{19} +26.8804 q^{21} -94.7913 q^{23} +250.455 q^{25} -144.106 q^{27} +29.0000 q^{29} +217.126 q^{31} -52.2873 q^{33} +153.634 q^{35} -183.461 q^{37} -154.558 q^{39} -13.4451 q^{41} -322.822 q^{43} -300.462 q^{45} -245.009 q^{47} -280.134 q^{49} -205.557 q^{51} -321.562 q^{53} -298.846 q^{55} -269.255 q^{57} +511.205 q^{59} -664.306 q^{61} -122.947 q^{63} -883.370 q^{65} +477.855 q^{67} -321.363 q^{69} -988.394 q^{71} -154.347 q^{73} +849.097 q^{75} -122.286 q^{77} -223.873 q^{79} -69.8789 q^{81} +1174.40 q^{83} -1174.85 q^{85} +98.3164 q^{87} +709.918 q^{89} -361.470 q^{91} +736.107 q^{93} -1538.91 q^{95} -802.826 q^{97} +239.155 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q + 4 q^{3} - 10 q^{5} - 12 q^{7} + 49 q^{9}+O(q^{10}) $ 9 * q + 4 * q^3 - 10 * q^5 - 12 * q^7 + 49 * q^9	$ 9 q + 4 q^{3} - 10 q^{5} - 12 q^{7} + 49 q^{9} + 64 q^{11} - 70 q^{13} - 170 q^{15} - 66 q^{17} + 42 q^{19} - 76 q^{21} - 40 q^{23} + 111 q^{25} + 322 q^{27} + 261 q^{29} + 64 q^{31} - 52 q^{33} + 496 q^{35} + 54 q^{37} - 590 q^{39} - 378 q^{41} - 32 q^{43} - 1046 q^{45} - 1164 q^{47} - 351 q^{49} + 376 q^{51} - 278 q^{53} - 614 q^{55} + 28 q^{57} + 640 q^{59} - 1054 q^{61} - 1660 q^{63} - 708 q^{65} + 1184 q^{67} - 188 q^{69} - 1988 q^{71} - 750 q^{73} + 3126 q^{75} - 1260 q^{77} - 2916 q^{79} + 293 q^{81} + 2832 q^{83} - 56 q^{85} + 116 q^{87} - 370 q^{89} + 3016 q^{91} + 1696 q^{93} - 4412 q^{95} - 2234 q^{97} + 4118 q^{99}+O(q^{100}) $ 9 * q + 4 * q^3 - 10 * q^5 - 12 * q^7 + 49 * q^9 + 64 * q^11 - 70 * q^13 - 170 * q^15 - 66 * q^17 + 42 * q^19 - 76 * q^21 - 40 * q^23 + 111 * q^25 + 322 * q^27 + 261 * q^29 + 64 * q^31 - 52 * q^33 + 496 * q^35 + 54 * q^37 - 590 * q^39 - 378 * q^41 - 32 * q^43 - 1046 * q^45 - 1164 * q^47 - 351 * q^49 + 376 * q^51 - 278 * q^53 - 614 * q^55 + 28 * q^57 + 640 * q^59 - 1054 * q^61 - 1660 * q^63 - 708 * q^65 + 1184 * q^67 - 188 * q^69 - 1988 * q^71 - 750 * q^73 + 3126 * q^75 - 1260 * q^77 - 2916 * q^79 + 293 * q^81 + 2832 * q^83 - 56 * q^85 + 116 * q^87 - 370 * q^89 + 3016 * q^91 + 1696 * q^93 - 4412 * q^95 - 2234 * q^97 + 4118 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	3.39022	0.652448	0.326224	−	0.945292i	$-0.394224\pi$
$3$	3.39022	0.652448	0.326224	+	0.945292i	$0.394224\pi$
$4$	0	0
$5$	19.3767	1.73310	0.866550	−	0.499090i	$-0.166332\pi$
$5$	19.3767	1.73310	0.866550	+	0.499090i	$0.166332\pi$
$6$	0	0
$7$	7.92881	0.428115	0.214058	−	0.976821i	$-0.431332\pi$
$7$	7.92881	0.428115	0.214058	+	0.976821i	$0.431332\pi$
$8$	0	0
$9$	−15.5064	−0.574311
$10$	0	0
$11$	−15.4230	−0.422745	−0.211373	−	0.977406i	$-0.567793\pi$
$11$	−15.4230	−0.422745	−0.211373	+	0.977406i	$0.567793\pi$
$12$	0	0
$13$	−45.5894	−0.972633	−0.486317	−	0.873783i	$-0.661660\pi$
$13$	−45.5894	−0.972633	−0.486317	+	0.873783i	$0.661660\pi$
$14$	0	0
$15$	65.6911	1.13076
$16$	0	0
$17$	−60.6323	−0.865028	−0.432514	−	0.901627i	$-0.642373\pi$
$17$	−60.6323	−0.865028	−0.432514	+	0.901627i	$0.642373\pi$
$18$	0	0
$19$	−79.4211	−0.958971	−0.479485	−	0.877550i	$-0.659177\pi$
$19$	−79.4211	−0.958971	−0.479485	+	0.877550i	$0.659177\pi$
$20$	0	0
$21$	26.8804	0.279323
$22$	0	0
$23$	−94.7913	−0.859363	−0.429681	−	0.902981i	$-0.641374\pi$
$23$	−94.7913	−0.859363	−0.429681	+	0.902981i	$0.641374\pi$
$24$	0	0
$25$	250.455	2.00364
$26$	0	0
$27$	−144.106	−1.02716
$28$	0	0
$29$	29.0000	0.185695
$29$	29.0000	0.185695
$30$	0	0
$31$	217.126	1.25797	0.628985	−	0.777417i	$-0.283471\pi$
$31$	217.126	1.25797	0.628985	+	0.777417i	$0.283471\pi$
$32$	0	0
$33$	−52.2873	−0.275820
$34$	0	0
$35$	153.634	0.741967
$36$	0	0
$37$	−183.461	−0.815157	−0.407578	−	0.913170i	$-0.633627\pi$
$37$	−183.461	−0.815157	−0.407578	+	0.913170i	$0.633627\pi$
$38$	0	0
$39$	−154.558	−0.634593
$40$	0	0
$41$	−13.4451	−0.0512141	−0.0256070	−	0.999672i	$-0.508152\pi$
$41$	−13.4451	−0.0512141	−0.0256070	+	0.999672i	$0.508152\pi$
$42$	0	0
$43$	−322.822	−1.14488	−0.572440	−	0.819947i	$-0.694003\pi$
$43$	−322.822	−1.14488	−0.572440	+	0.819947i	$0.694003\pi$
$44$	0	0
$45$	−300.462	−0.995339
$46$	0	0
$47$	−245.009	−0.760387	−0.380194	−	0.924907i	$-0.624143\pi$
$47$	−245.009	−0.760387	−0.380194	+	0.924907i	$0.624143\pi$
$48$	0	0
$49$	−280.134	−0.816717
$50$	0	0
$51$	−205.557	−0.564386
$52$	0	0
$53$	−321.562	−0.833396	−0.416698	−	0.909045i	$-0.636813\pi$
$53$	−321.562	−0.833396	−0.416698	+	0.909045i	$0.636813\pi$
$54$	0	0
$55$	−298.846	−0.732660
$56$	0	0
$57$	−269.255	−0.625679
$58$	0	0
$59$	511.205	1.12802	0.564010	−	0.825768i	$-0.309258\pi$
$59$	511.205	1.12802	0.564010	+	0.825768i	$0.309258\pi$
$60$	0	0
$61$	−664.306	−1.39435	−0.697177	−	0.716899i	$-0.745561\pi$
$61$	−664.306	−1.39435	−0.697177	+	0.716899i	$0.745561\pi$
$62$	0	0
$63$	−122.947	−0.245871
$64$	0	0
$65$	−883.370	−1.68567
$66$	0	0
$67$	477.855	0.871333	0.435667	−	0.900108i	$-0.356513\pi$
$67$	477.855	0.871333	0.435667	+	0.900108i	$0.356513\pi$
$68$	0	0
$69$	−321.363	−0.560690
$70$	0	0
$71$	−988.394	−1.65212	−0.826062	−	0.563580i	$-0.809424\pi$
$71$	−988.394	−1.65212	−0.826062	+	0.563580i	$0.809424\pi$
$72$	0	0
$73$	−154.347	−0.247466	−0.123733	−	0.992316i	$-0.539487\pi$
$73$	−154.347	−0.247466	−0.123733	+	0.992316i	$0.539487\pi$
$74$	0	0
$75$	849.097	1.30727
$76$	0	0
$77$	−122.286	−0.180984
$78$	0	0
$79$	−223.873	−0.318831	−0.159415	−	0.987212i	$-0.550961\pi$
$79$	−223.873	−0.318831	−0.159415	+	0.987212i	$0.550961\pi$
$80$	0	0
$81$	−69.8789	−0.0958559
$82$	0	0
$83$	1174.40	1.55310	0.776552	−	0.630053i	$-0.216967\pi$
$83$	1174.40	1.55310	0.776552	+	0.630053i	$0.216967\pi$
$84$	0	0
$85$	−1174.85	−1.49918
$86$	0	0
$87$	98.3164	0.121157
$88$	0	0
$89$	709.918	0.845519	0.422759	−	0.906242i	$-0.361062\pi$
$89$	709.918	0.845519	0.422759	+	0.906242i	$0.361062\pi$
$90$	0	0
$91$	−361.470	−0.416399
$92$	0	0
$93$	736.107	0.820761
$94$	0	0
$95$	−1538.91	−1.66199
$96$	0	0
$97$	−802.826	−0.840357	−0.420178	−	0.907441i	$-0.638033\pi$
$97$	−802.826	−0.840357	−0.420178	+	0.907441i	$0.638033\pi$
$98$	0	0
$99$	239.155	0.242787
$100$	0	0
$101$	−696.175	−0.685861	−0.342931	−	0.939361i	$-0.611420\pi$
$101$	−696.175	−0.685861	−0.342931	+	0.939361i	$0.611420\pi$
$102$	0	0
$103$	225.517	0.215736	0.107868	−	0.994165i	$-0.465598\pi$
$103$	225.517	0.215736	0.107868	+	0.994165i	$0.465598\pi$
$104$	0	0
$105$	520.852	0.484095
$106$	0	0
$107$	−754.805	−0.681960	−0.340980	−	0.940071i	$-0.610759\pi$
$107$	−754.805	−0.681960	−0.340980	+	0.940071i	$0.610759\pi$
$108$	0	0
$109$	1877.63	1.64995	0.824974	−	0.565170i	$-0.191190\pi$
$109$	1877.63	1.64995	0.824974	+	0.565170i	$0.191190\pi$
$110$	0	0
$111$	−621.973	−0.531848
$112$	0	0
$113$	478.920	0.398699	0.199349	−	0.979928i	$-0.436117\pi$
$113$	478.920	0.398699	0.199349	+	0.979928i	$0.436117\pi$
$114$	0	0
$115$	−1836.74	−1.48936
$116$	0	0
$117$	706.928	0.558594
$118$	0	0
$119$	−480.741	−0.370332
$120$	0	0
$121$	−1093.13	−0.821286
$122$	0	0
$123$	−45.5820	−0.0334146
$124$	0	0
$125$	2430.89	1.73940
$126$	0	0
$127$	−1178.05	−0.823112	−0.411556	−	0.911384i	$-0.635015\pi$
$127$	−1178.05	−0.823112	−0.411556	+	0.911384i	$0.635015\pi$
$128$	0	0
$129$	−1094.44	−0.746975
$130$	0	0
$131$	1363.44	0.909347	0.454673	−	0.890658i	$-0.349756\pi$
$131$	1363.44	0.909347	0.454673	+	0.890658i	$0.349756\pi$
$132$	0	0
$133$	−629.714	−0.410550
$134$	0	0
$135$	−2792.29	−1.78017
$136$	0	0
$137$	−709.000	−0.442145	−0.221073	−	0.975257i	$-0.570956\pi$
$137$	−709.000	−0.442145	−0.221073	+	0.975257i	$0.570956\pi$
$138$	0	0
$139$	2088.68	1.27453	0.637264	−	0.770645i	$-0.280066\pi$
$139$	2088.68	1.27453	0.637264	+	0.770645i	$0.280066\pi$
$140$	0	0
$141$	−830.634	−0.496113
$142$	0	0
$143$	703.124	0.411176
$144$	0	0
$145$	561.923	0.321829
$146$	0	0
$147$	−949.716	−0.532866
$148$	0	0
$149$	−3552.78	−1.95339	−0.976696	−	0.214627i	$-0.931147\pi$
$149$	−3552.78	−1.95339	−0.976696	+	0.214627i	$0.931147\pi$
$150$	0	0
$151$	1263.48	0.680931	0.340465	−	0.940257i	$-0.389415\pi$
$151$	1263.48	0.680931	0.340465	+	0.940257i	$0.389415\pi$
$152$	0	0
$153$	940.188	0.496795
$154$	0	0
$155$	4207.18	2.18019
$156$	0	0
$157$	2205.44	1.12110	0.560551	−	0.828120i	$-0.310589\pi$
$157$	2205.44	1.12110	0.560551	+	0.828120i	$0.310589\pi$
$158$	0	0
$159$	−1090.17	−0.543748
$160$	0	0
$161$	−751.582	−0.367906
$162$	0	0
$163$	322.051	0.154754	0.0773771	−	0.997002i	$-0.475345\pi$
$163$	322.051	0.154754	0.0773771	+	0.997002i	$0.475345\pi$
$164$	0	0
$165$	−1013.15	−0.478023
$166$	0	0
$167$	4074.72	1.88809	0.944046	−	0.329815i	$-0.106986\pi$
$167$	4074.72	1.88809	0.944046	+	0.329815i	$0.106986\pi$
$168$	0	0
$169$	−118.605	−0.0539848
$170$	0	0
$171$	1231.53	0.550747
$172$	0	0
$173$	171.789	0.0754964	0.0377482	−	0.999287i	$-0.487982\pi$
$173$	171.789	0.0754964	0.0377482	+	0.999287i	$0.487982\pi$
$174$	0	0
$175$	1985.81	0.857788
$176$	0	0
$177$	1733.10	0.735975
$178$	0	0
$179$	−2612.73	−1.09098	−0.545489	−	0.838118i	$-0.683656\pi$
$179$	−2612.73	−1.09098	−0.545489	+	0.838118i	$0.683656\pi$
$180$	0	0
$181$	1084.07	0.445184	0.222592	−	0.974912i	$-0.428548\pi$
$181$	1084.07	0.445184	0.222592	+	0.974912i	$0.428548\pi$
$182$	0	0
$183$	−2252.14	−0.909745
$184$	0	0
$185$	−3554.86	−1.41275
$186$	0	0
$187$	935.129	0.365687
$188$	0	0
$189$	−1142.59	−0.439742
$190$	0	0
$191$	2499.34	0.946839	0.473419	−	0.880837i	$-0.343020\pi$
$191$	2499.34	0.946839	0.473419	+	0.880837i	$0.343020\pi$
$192$	0	0
$193$	−1189.90	−0.443785	−0.221893	−	0.975071i	$-0.571223\pi$
$193$	−1189.90	−0.443785	−0.221893	+	0.975071i	$0.571223\pi$
$194$	0	0
$195$	−2994.82	−1.09981
$196$	0	0
$197$	3048.06	1.10236	0.551181	−	0.834386i	$-0.314177\pi$
$197$	3048.06	1.10236	0.551181	+	0.834386i	$0.314177\pi$
$198$	0	0
$199$	−1282.94	−0.457009	−0.228505	−	0.973543i	$-0.573384\pi$
$199$	−1282.94	−0.457009	−0.228505	+	0.973543i	$0.573384\pi$
$200$	0	0
$201$	1620.04	0.568500
$202$	0	0
$203$	229.935	0.0794990
$204$	0	0
$205$	−260.522	−0.0887592
$206$	0	0
$207$	1469.87	0.493542
$208$	0	0
$209$	1224.91	0.405401
$210$	0	0
$211$	−744.561	−0.242927	−0.121464	−	0.992596i	$-0.538759\pi$
$211$	−744.561	−0.242927	−0.121464	+	0.992596i	$0.538759\pi$
$212$	0	0
$213$	−3350.87	−1.07793
$214$	0	0
$215$	−6255.20	−1.98419
$216$	0	0
$217$	1721.55	0.538556
$218$	0	0
$219$	−523.272	−0.161459
$220$	0	0
$221$	2764.19	0.841355
$222$	0	0
$223$	770.021	0.231231	0.115615	−	0.993294i	$-0.463116\pi$
$223$	770.021	0.231231	0.115615	+	0.993294i	$0.463116\pi$
$224$	0	0
$225$	−3883.65	−1.15071
$226$	0	0
$227$	5567.58	1.62790	0.813950	−	0.580935i	$-0.197313\pi$
$227$	5567.58	1.62790	0.813950	+	0.580935i	$0.197313\pi$
$228$	0	0
$229$	−4524.04	−1.30549	−0.652744	−	0.757578i	$-0.726382\pi$
$229$	−4524.04	−1.30549	−0.652744	+	0.757578i	$0.726382\pi$
$230$	0	0
$231$	−414.576	−0.118083
$232$	0	0
$233$	1448.90	0.407383	0.203692	−	0.979035i	$-0.434706\pi$
$233$	1448.90	0.407383	0.203692	+	0.979035i	$0.434706\pi$
$234$	0	0
$235$	−4747.45	−1.31783
$236$	0	0
$237$	−758.978	−0.208021
$238$	0	0
$239$	−3964.90	−1.07309	−0.536544	−	0.843873i	$-0.680270\pi$
$239$	−3964.90	−1.07309	−0.536544	+	0.843873i	$0.680270\pi$
$240$	0	0
$241$	2278.49	0.609006	0.304503	−	0.952511i	$-0.401510\pi$
$241$	2278.49	0.609006	0.304503	+	0.952511i	$0.401510\pi$
$242$	0	0
$243$	3653.96	0.964616
$244$	0	0
$245$	−5428.06	−1.41545
$246$	0	0
$247$	3620.76	0.932727
$248$	0	0
$249$	3981.49	1.01332
$250$	0	0
$251$	4850.92	1.21987	0.609935	−	0.792451i	$-0.291195\pi$
$251$	4850.92	1.21987	0.609935	+	0.792451i	$0.291195\pi$
$252$	0	0
$253$	1461.96	0.363292
$254$	0	0
$255$	−3983.00	−0.978138
$256$	0	0
$257$	−7106.13	−1.72478	−0.862390	−	0.506244i	$-0.831033\pi$
$257$	−7106.13	−1.72478	−0.862390	+	0.506244i	$0.831033\pi$
$258$	0	0
$259$	−1454.63	−0.348981
$260$	0	0
$261$	−449.686	−0.106647
$262$	0	0
$263$	−2543.06	−0.596242	−0.298121	−	0.954528i	$-0.596360\pi$
$263$	−2543.06	−0.596242	−0.298121	+	0.954528i	$0.596360\pi$
$264$	0	0
$265$	−6230.80	−1.44436
$266$	0	0
$267$	2406.78	0.551657
$268$	0	0
$269$	−7413.71	−1.68038	−0.840190	−	0.542293i	$-0.817556\pi$
$269$	−7413.71	−1.68038	−0.840190	+	0.542293i	$0.817556\pi$
$270$	0	0
$271$	−618.554	−0.138651	−0.0693256	−	0.997594i	$-0.522085\pi$
$271$	−618.554	−0.138651	−0.0693256	+	0.997594i	$0.522085\pi$
$272$	0	0
$273$	−1225.46	−0.271679
$274$	0	0
$275$	−3862.75	−0.847029
$276$	0	0
$277$	3471.99	0.753110	0.376555	−	0.926394i	$-0.377109\pi$
$277$	3471.99	0.753110	0.376555	+	0.926394i	$0.377109\pi$
$278$	0	0
$279$	−3366.85	−0.722466
$280$	0	0
$281$	−4494.51	−0.954163	−0.477082	−	0.878859i	$-0.658305\pi$
$281$	−4494.51	−0.954163	−0.477082	+	0.878859i	$0.658305\pi$
$282$	0	0
$283$	2869.00	0.602631	0.301316	−	0.953524i	$-0.402574\pi$
$283$	2869.00	0.602631	0.301316	+	0.953524i	$0.402574\pi$
$284$	0	0
$285$	−5217.26	−1.08436
$286$	0	0
$287$	−106.604	−0.0219255
$288$	0	0
$289$	−1236.73	−0.251726
$290$	0	0
$291$	−2721.76	−0.548290
$292$	0	0
$293$	−5463.18	−1.08929	−0.544646	−	0.838666i	$-0.683336\pi$
$293$	−5463.18	−1.08929	−0.544646	+	0.838666i	$0.683336\pi$
$294$	0	0
$295$	9905.43	1.95497
$296$	0	0
$297$	2222.54	0.434226
$298$	0	0
$299$	4321.48	0.835845
$300$	0	0
$301$	−2559.59	−0.490141
$302$	0	0
$303$	−2360.19	−0.447489
$304$	0	0
$305$	−12872.0	−2.41656
$306$	0	0
$307$	6160.56	1.14528	0.572641	−	0.819806i	$-0.305919\pi$
$307$	6160.56	1.14528	0.572641	+	0.819806i	$0.305919\pi$
$308$	0	0
$309$	764.552	0.140757
$310$	0	0
$311$	−1676.59	−0.305693	−0.152846	−	0.988250i	$-0.548844\pi$
$311$	−1676.59	−0.305693	−0.152846	+	0.988250i	$0.548844\pi$
$312$	0	0
$313$	2401.33	0.433647	0.216823	−	0.976211i	$-0.430430\pi$
$313$	2401.33	0.433647	0.216823	+	0.976211i	$0.430430\pi$
$314$	0	0
$315$	−2382.31	−0.426120
$316$	0	0
$317$	−5693.87	−1.00883	−0.504415	−	0.863461i	$-0.668292\pi$
$317$	−5693.87	−1.00883	−0.504415	+	0.863461i	$0.668292\pi$
$318$	0	0
$319$	−447.266	−0.0785019
$320$	0	0
$321$	−2558.95	−0.444944
$322$	0	0
$323$	4815.48	0.829537
$324$	0	0
$325$	−11418.1	−1.94880
$326$	0	0
$327$	6365.58	1.07651
$328$	0	0
$329$	−1942.63	−0.325533
$330$	0	0
$331$	−3478.46	−0.577624	−0.288812	−	0.957386i	$-0.593260\pi$
$331$	−3478.46	−0.577624	−0.288812	+	0.957386i	$0.593260\pi$
$332$	0	0
$333$	2844.82	0.468153
$334$	0	0
$335$	9259.24	1.51011
$336$	0	0
$337$	7228.24	1.16839	0.584195	−	0.811614i	$-0.301410\pi$
$337$	7228.24	1.16839	0.584195	+	0.811614i	$0.301410\pi$
$338$	0	0
$339$	1623.64	0.260130
$340$	0	0
$341$	−3348.74	−0.531801
$342$	0	0
$343$	−4940.71	−0.777765
$344$	0	0
$345$	−6226.95	−0.971732
$346$	0	0
$347$	−7713.95	−1.19339	−0.596695	−	0.802468i	$-0.703520\pi$
$347$	−7713.95	−1.19339	−0.596695	+	0.802468i	$0.703520\pi$
$348$	0	0
$349$	9763.60	1.49752	0.748758	−	0.662843i	$-0.230650\pi$
$349$	9763.60	1.49752	0.748758	+	0.662843i	$0.230650\pi$
$350$	0	0
$351$	6569.71	0.999047
$352$	0	0
$353$	−4230.58	−0.637878	−0.318939	−	0.947775i	$-0.603326\pi$
$353$	−4230.58	−0.637878	−0.318939	+	0.947775i	$0.603326\pi$
$354$	0	0
$355$	−19151.8	−2.86330
$356$	0	0
$357$	−1629.82	−0.241622
$358$	0	0
$359$	5847.37	0.859645	0.429823	−	0.902913i	$-0.358576\pi$
$359$	5847.37	0.859645	0.429823	+	0.902913i	$0.358576\pi$
$360$	0	0
$361$	−551.292	−0.0803750
$362$	0	0
$363$	−3705.96	−0.535847
$364$	0	0
$365$	−2990.74	−0.428883
$366$	0	0
$367$	9915.51	1.41031	0.705157	−	0.709051i	$-0.250876\pi$
$367$	9915.51	1.41031	0.705157	+	0.709051i	$0.250876\pi$
$368$	0	0
$369$	208.486	0.0294128
$370$	0	0
$371$	−2549.61	−0.356790
$372$	0	0
$373$	2758.32	0.382897	0.191448	−	0.981503i	$-0.438682\pi$
$373$	2758.32	0.382897	0.191448	+	0.981503i	$0.438682\pi$
$374$	0	0
$375$	8241.26	1.13487
$376$	0	0
$377$	−1322.09	−0.180613
$378$	0	0
$379$	5044.05	0.683630	0.341815	−	0.939767i	$-0.388958\pi$
$379$	5044.05	0.683630	0.341815	+	0.939767i	$0.388958\pi$
$380$	0	0
$381$	−3993.86	−0.537038
$382$	0	0
$383$	3039.31	0.405487	0.202744	−	0.979232i	$-0.435014\pi$
$383$	3039.31	0.405487	0.202744	+	0.979232i	$0.435014\pi$
$384$	0	0
$385$	−2369.49	−0.313663
$386$	0	0
$387$	5005.80	0.657517
$388$	0	0
$389$	−5131.66	−0.668857	−0.334428	−	0.942421i	$-0.608543\pi$
$389$	−5131.66	−0.668857	−0.334428	+	0.942421i	$0.608543\pi$
$390$	0	0
$391$	5747.41	0.743373
$392$	0	0
$393$	4622.37	0.593302
$394$	0	0
$395$	−4337.90	−0.552566
$396$	0	0
$397$	−3554.10	−0.449308	−0.224654	−	0.974439i	$-0.572125\pi$
$397$	−3554.10	−0.449308	−0.224654	+	0.974439i	$0.572125\pi$
$398$	0	0
$399$	−2134.87	−0.267863
$400$	0	0
$401$	−4176.12	−0.520063	−0.260032	−	0.965600i	$-0.583733\pi$
$401$	−4176.12	−0.520063	−0.260032	+	0.965600i	$0.583733\pi$
$402$	0	0
$403$	−9898.67	−1.22354
$404$	0	0
$405$	−1354.02	−0.166128
$406$	0	0
$407$	2829.51	0.344604
$408$	0	0
$409$	−8769.07	−1.06015	−0.530077	−	0.847950i	$-0.677837\pi$
$409$	−8769.07	−1.06015	−0.530077	+	0.847950i	$0.677837\pi$
$410$	0	0
$411$	−2403.67	−0.288477
$412$	0	0
$413$	4053.24	0.482923
$414$	0	0
$415$	22756.0	2.69168
$416$	0	0
$417$	7081.09	0.831564
$418$	0	0
$419$	12483.9	1.45556	0.727781	−	0.685810i	$-0.240552\pi$
$419$	12483.9	1.45556	0.727781	+	0.685810i	$0.240552\pi$
$420$	0	0
$421$	−5644.50	−0.653435	−0.326717	−	0.945122i	$-0.605942\pi$
$421$	−5644.50	−0.653435	−0.326717	+	0.945122i	$0.605942\pi$
$422$	0	0
$423$	3799.20	0.436699
$424$	0	0
$425$	−15185.6	−1.73320
$426$	0	0
$427$	−5267.15	−0.596945
$428$	0	0
$429$	2383.75	0.268271
$430$	0	0
$431$	−3285.43	−0.367178	−0.183589	−	0.983003i	$-0.558771\pi$
$431$	−3285.43	−0.367178	−0.183589	+	0.983003i	$0.558771\pi$
$432$	0	0
$433$	−2948.21	−0.327210	−0.163605	−	0.986526i	$-0.552312\pi$
$433$	−2948.21	−0.327210	−0.163605	+	0.986526i	$0.552312\pi$
$434$	0	0
$435$	1905.04	0.209977
$436$	0	0
$437$	7528.42	0.824104
$438$	0	0
$439$	−17019.6	−1.85034	−0.925170	−	0.379553i	$-0.876078\pi$
$439$	−17019.6	−1.85034	−0.925170	+	0.379553i	$0.876078\pi$
$440$	0	0
$441$	4343.87	0.469050
$442$	0	0
$443$	−13817.4	−1.48191	−0.740954	−	0.671556i	$-0.765626\pi$
$443$	−13817.4	−1.48191	−0.740954	+	0.671556i	$0.765626\pi$
$444$	0	0
$445$	13755.8	1.46537
$446$	0	0
$447$	−12044.7	−1.27449
$448$	0	0
$449$	−5738.26	−0.603129	−0.301565	−	0.953446i	$-0.597509\pi$
$449$	−5738.26	−0.603129	−0.301565	+	0.953446i	$0.597509\pi$
$450$	0	0
$451$	207.364	0.0216505
$452$	0	0
$453$	4283.48	0.444272
$454$	0	0
$455$	−7004.07	−0.721662
$456$	0	0
$457$	8452.49	0.865188	0.432594	−	0.901589i	$-0.357598\pi$
$457$	8452.49	0.865188	0.432594	+	0.901589i	$0.357598\pi$
$458$	0	0
$459$	8737.48	0.888520
$460$	0	0
$461$	−7824.62	−0.790518	−0.395259	−	0.918570i	$-0.629345\pi$
$461$	−7824.62	−0.790518	−0.395259	+	0.918570i	$0.629345\pi$
$462$	0	0
$463$	370.241	0.0371632	0.0185816	−	0.999827i	$-0.494085\pi$
$463$	370.241	0.0371632	0.0185816	+	0.999827i	$0.494085\pi$
$464$	0	0
$465$	14263.3	1.42246
$466$	0	0
$467$	−17808.8	−1.76466	−0.882329	−	0.470634i	$-0.844025\pi$
$467$	−17808.8	−1.76466	−0.882329	+	0.470634i	$0.844025\pi$
$468$	0	0
$469$	3788.82	0.373031
$470$	0	0
$471$	7476.92	0.731462
$472$	0	0
$473$	4978.87	0.483993
$474$	0	0
$475$	−19891.4	−1.92143
$476$	0	0
$477$	4986.28	0.478629
$478$	0	0
$479$	9586.99	0.914490	0.457245	−	0.889341i	$-0.348836\pi$
$479$	9586.99	0.914490	0.457245	+	0.889341i	$0.348836\pi$
$480$	0	0
$481$	8363.88	0.792848
$482$	0	0
$483$	−2548.03	−0.240040
$484$	0	0
$485$	−15556.1	−1.45642
$486$	0	0
$487$	−17317.1	−1.61132	−0.805661	−	0.592377i	$-0.798190\pi$
$487$	−17317.1	−1.61132	−0.805661	+	0.592377i	$0.798190\pi$
$488$	0	0
$489$	1091.82	0.100969
$490$	0	0
$491$	16160.0	1.48532	0.742658	−	0.669671i	$-0.233565\pi$
$491$	16160.0	1.48532	0.742658	+	0.669671i	$0.233565\pi$
$492$	0	0
$493$	−1758.34	−0.160632
$494$	0	0
$495$	4634.02	0.420775
$496$	0	0
$497$	−7836.78	−0.707299
$498$	0	0
$499$	−11711.1	−1.05063	−0.525313	−	0.850909i	$-0.676052\pi$
$499$	−11711.1	−1.05063	−0.525313	+	0.850909i	$0.676052\pi$
$500$	0	0
$501$	13814.2	1.23188
$502$	0	0
$503$	−6140.30	−0.544299	−0.272150	−	0.962255i	$-0.587735\pi$
$503$	−6140.30	−0.544299	−0.272150	+	0.962255i	$0.587735\pi$
$504$	0	0
$505$	−13489.5	−1.18867
$506$	0	0
$507$	−402.096	−0.0352223
$508$	0	0
$509$	4198.30	0.365592	0.182796	−	0.983151i	$-0.441485\pi$
$509$	4198.30	0.365592	0.182796	+	0.983151i	$0.441485\pi$
$510$	0	0
$511$	−1223.79	−0.105944
$512$	0	0
$513$	11445.1	0.985013
$514$	0	0
$515$	4369.76	0.373893
$516$	0	0
$517$	3778.76	0.321450
$518$	0	0
$519$	582.403	0.0492575
$520$	0	0
$521$	−4625.41	−0.388950	−0.194475	−	0.980908i	$-0.562300\pi$
$521$	−4625.41	−0.388950	−0.194475	+	0.980908i	$0.562300\pi$
$522$	0	0
$523$	−6063.22	−0.506933	−0.253467	−	0.967344i	$-0.581571\pi$
$523$	−6063.22	−0.506933	−0.253467	+	0.967344i	$0.581571\pi$
$524$	0	0
$525$	6732.33	0.559662
$526$	0	0
$527$	−13164.9	−1.08818
$528$	0	0
$529$	−3181.62	−0.261496
$530$	0	0
$531$	−7926.94	−0.647834
$532$	0	0
$533$	612.956	0.0498125
$534$	0	0
$535$	−14625.6	−1.18191
$536$	0	0
$537$	−8857.75	−0.711806
$538$	0	0
$539$	4320.50	0.345264
$540$	0	0
$541$	6749.03	0.536347	0.268173	−	0.963371i	$-0.413580\pi$
$541$	6749.03	0.536347	0.268173	+	0.963371i	$0.413580\pi$
$542$	0	0
$543$	3675.24	0.290460
$544$	0	0
$545$	36382.2	2.85953
$546$	0	0
$547$	2593.58	0.202730	0.101365	−	0.994849i	$-0.467679\pi$
$547$	2593.58	0.202730	0.101365	+	0.994849i	$0.467679\pi$
$548$	0	0
$549$	10301.0	0.800793
$550$	0	0
$551$	−2303.21	−0.178076
$552$	0	0
$553$	−1775.04	−0.136496
$554$	0	0
$555$	−12051.8	−0.921745
$556$	0	0
$557$	17018.7	1.29463	0.647313	−	0.762224i	$-0.275893\pi$
$557$	17018.7	1.29463	0.647313	+	0.762224i	$0.275893\pi$
$558$	0	0
$559$	14717.3	1.11355
$560$	0	0
$561$	3170.30	0.238592
$562$	0	0
$563$	17896.0	1.33966	0.669830	−	0.742515i	$-0.266367\pi$
$563$	17896.0	1.33966	0.669830	+	0.742515i	$0.266367\pi$
$564$	0	0
$565$	9279.86	0.690985
$566$	0	0
$567$	−554.057	−0.0410374
$568$	0	0
$569$	−2825.90	−0.208204	−0.104102	−	0.994567i	$-0.533197\pi$
$569$	−2825.90	−0.208204	−0.104102	+	0.994567i	$0.533197\pi$
$570$	0	0
$571$	23622.5	1.73130	0.865650	−	0.500650i	$-0.166906\pi$
$571$	23622.5	1.73130	0.865650	+	0.500650i	$0.166906\pi$
$572$	0	0
$573$	8473.33	0.617764
$574$	0	0
$575$	−23740.9	−1.72185
$576$	0	0
$577$	−9005.48	−0.649745	−0.324873	−	0.945758i	$-0.605321\pi$
$577$	−9005.48	−0.649745	−0.324873	+	0.945758i	$0.605321\pi$
$578$	0	0
$579$	−4034.01	−0.289547
$580$	0	0
$581$	9311.63	0.664908
$582$	0	0
$583$	4959.45	0.352315
$584$	0	0
$585$	13697.9	0.968099
$586$	0	0
$587$	13008.8	0.914705	0.457353	−	0.889285i	$-0.348798\pi$
$587$	13008.8	0.914705	0.457353	+	0.889285i	$0.348798\pi$
$588$	0	0
$589$	−17244.4	−1.20636
$590$	0	0
$591$	10333.6	0.719234
$592$	0	0
$593$	9331.77	0.646222	0.323111	−	0.946361i	$-0.395271\pi$
$593$	9331.77	0.646222	0.323111	+	0.946361i	$0.395271\pi$
$594$	0	0
$595$	−9315.16	−0.641822
$596$	0	0
$597$	−4349.44	−0.298175
$598$	0	0
$599$	11051.3	0.753831	0.376916	−	0.926248i	$-0.376985\pi$
$599$	11051.3	0.753831	0.376916	+	0.926248i	$0.376985\pi$
$600$	0	0
$601$	−1083.67	−0.0735503	−0.0367752	−	0.999324i	$-0.511709\pi$
$601$	−1083.67	−0.0735503	−0.0367752	+	0.999324i	$0.511709\pi$
$602$	0	0
$603$	−7409.81	−0.500416
$604$	0	0
$605$	−21181.2	−1.42337
$606$	0	0
$607$	−20189.0	−1.34999	−0.674997	−	0.737820i	$-0.735855\pi$
$607$	−20189.0	−1.34999	−0.674997	+	0.737820i	$0.735855\pi$
$608$	0	0
$609$	779.532	0.0518690
$610$	0	0
$611$	11169.8	0.739578
$612$	0	0
$613$	−25164.1	−1.65802	−0.829012	−	0.559231i	$-0.811096\pi$
$613$	−25164.1	−1.65802	−0.829012	+	0.559231i	$0.811096\pi$
$614$	0	0
$615$	−883.226	−0.0579108
$616$	0	0
$617$	8089.78	0.527848	0.263924	−	0.964543i	$-0.414983\pi$
$617$	8089.78	0.527848	0.263924	+	0.964543i	$0.414983\pi$
$618$	0	0
$619$	26830.2	1.74216	0.871081	−	0.491139i	$-0.163419\pi$
$619$	26830.2	1.74216	0.871081	+	0.491139i	$0.163419\pi$
$620$	0	0
$621$	13660.0	0.882700
$622$	0	0
$623$	5628.80	0.361980
$624$	0	0
$625$	15795.7	1.01093
$626$	0	0
$627$	4152.71	0.264503
$628$	0	0
$629$	11123.7	0.705134
$630$	0	0
$631$	−23534.6	−1.48478	−0.742392	−	0.669965i	$-0.766309\pi$
$631$	−23534.6	−1.48478	−0.742392	+	0.669965i	$0.766309\pi$
$632$	0	0
$633$	−2524.23	−0.158498
$634$	0	0
$635$	−22826.7	−1.42654
$636$	0	0
$637$	12771.1	0.794366
$638$	0	0
$639$	15326.4	0.948833
$640$	0	0
$641$	26058.4	1.60568	0.802842	−	0.596192i	$-0.203320\pi$
$641$	26058.4	1.60568	0.802842	+	0.596192i	$0.203320\pi$
$642$	0	0
$643$	6498.37	0.398555	0.199277	−	0.979943i	$-0.436141\pi$
$643$	6498.37	0.398555	0.199277	+	0.979943i	$0.436141\pi$
$644$	0	0
$645$	−21206.5	−1.29458
$646$	0	0
$647$	−7402.89	−0.449826	−0.224913	−	0.974379i	$-0.572210\pi$
$647$	−7402.89	−0.449826	−0.224913	+	0.974379i	$0.572210\pi$
$648$	0	0
$649$	−7884.29	−0.476865
$650$	0	0
$651$	5836.45	0.351380
$652$	0	0
$653$	−19401.4	−1.16269	−0.581344	−	0.813658i	$-0.697473\pi$
$653$	−19401.4	−1.16269	−0.581344	+	0.813658i	$0.697473\pi$
$654$	0	0
$655$	26418.9	1.57599
$656$	0	0
$657$	2393.37	0.142122
$658$	0	0
$659$	−15564.9	−0.920067	−0.460033	−	0.887902i	$-0.652163\pi$
$659$	−15564.9	−0.920067	−0.460033	+	0.887902i	$0.652163\pi$
$660$	0	0
$661$	849.108	0.0499644	0.0249822	−	0.999688i	$-0.492047\pi$
$661$	849.108	0.0499644	0.0249822	+	0.999688i	$0.492047\pi$
$662$	0	0
$663$	9371.21	0.548941
$664$	0	0
$665$	−12201.8	−0.711525
$666$	0	0
$667$	−2748.95	−0.159580
$668$	0	0
$669$	2610.54	0.150866
$670$	0	0
$671$	10245.6	0.589457
$672$	0	0
$673$	1970.54	0.112866	0.0564328	−	0.998406i	$-0.482027\pi$
$673$	1970.54	0.112866	0.0564328	+	0.998406i	$0.482027\pi$
$674$	0	0
$675$	−36092.0	−2.05805
$676$	0	0
$677$	226.339	0.0128492	0.00642460	−	0.999979i	$-0.497955\pi$
$677$	226.339	0.0128492	0.00642460	+	0.999979i	$0.497955\pi$
$678$	0	0
$679$	−6365.45	−0.359770
$680$	0	0
$681$	18875.3	1.06212
$682$	0	0
$683$	−2850.00	−0.159667	−0.0798333	−	0.996808i	$-0.525439\pi$
$683$	−2850.00	−0.159667	−0.0798333	+	0.996808i	$0.525439\pi$
$684$	0	0
$685$	−13738.0	−0.766283
$686$	0	0
$687$	−15337.5	−0.851764
$688$	0	0
$689$	14659.8	0.810589
$690$	0	0
$691$	−8042.88	−0.442787	−0.221393	−	0.975185i	$-0.571060\pi$
$691$	−8042.88	−0.442787	−0.221393	+	0.975185i	$0.571060\pi$
$692$	0	0
$693$	1896.21	0.103941
$694$	0	0
$695$	40471.6	2.20889
$696$	0	0
$697$	815.209	0.0443016
$698$	0	0
$699$	4912.08	0.265797
$700$	0	0
$701$	6565.07	0.353722	0.176861	−	0.984236i	$-0.443406\pi$
$701$	6565.07	0.353722	0.176861	+	0.984236i	$0.443406\pi$
$702$	0	0
$703$	14570.7	0.781711
$704$	0	0
$705$	−16094.9	−0.859815
$706$	0	0
$707$	−5519.84	−0.293628
$708$	0	0
$709$	−3420.82	−0.181201	−0.0906004	−	0.995887i	$-0.528879\pi$
$709$	−3420.82	−0.181201	−0.0906004	+	0.995887i	$0.528879\pi$
$710$	0	0
$711$	3471.46	0.183108
$712$	0	0
$713$	−20581.7	−1.08105
$714$	0	0
$715$	13624.2	0.712610
$716$	0	0
$717$	−13441.9	−0.700134
$718$	0	0
$719$	−33111.5	−1.71746	−0.858729	−	0.512430i	$-0.828745\pi$
$719$	−33111.5	−1.71746	−0.858729	+	0.512430i	$0.828745\pi$
$720$	0	0
$721$	1788.08	0.0923600
$722$	0	0
$723$	7724.59	0.397345
$724$	0	0
$725$	7263.19	0.372066
$726$	0	0
$727$	−15639.6	−0.797857	−0.398928	−	0.916982i	$-0.630618\pi$
$727$	−15639.6	−0.797857	−0.398928	+	0.916982i	$0.630618\pi$
$728$	0	0
$729$	14274.5	0.725218
$730$	0	0
$731$	19573.4	0.990354
$732$	0	0
$733$	15822.7	0.797303	0.398652	−	0.917102i	$-0.369478\pi$
$733$	15822.7	0.797303	0.398652	+	0.917102i	$0.369478\pi$
$734$	0	0
$735$	−18402.3	−0.923510
$736$	0	0
$737$	−7369.95	−0.368352
$738$	0	0
$739$	12462.7	0.620360	0.310180	−	0.950678i	$-0.399611\pi$
$739$	12462.7	0.620360	0.310180	+	0.950678i	$0.399611\pi$
$740$	0	0
$741$	12275.2	0.608556
$742$	0	0
$743$	−33179.8	−1.63829	−0.819144	−	0.573588i	$-0.805551\pi$
$743$	−33179.8	−1.63829	−0.819144	+	0.573588i	$0.805551\pi$
$744$	0	0
$745$	−68841.1	−3.38543
$746$	0	0
$747$	−18210.8	−0.891965
$748$	0	0
$749$	−5984.70	−0.291958
$750$	0	0
$751$	−26007.0	−1.26366	−0.631829	−	0.775108i	$-0.717696\pi$
$751$	−26007.0	−1.26366	−0.631829	+	0.775108i	$0.717696\pi$
$752$	0	0
$753$	16445.7	0.795903
$754$	0	0
$755$	24482.0	1.18012
$756$	0	0
$757$	35312.0	1.69543	0.847713	−	0.530455i	$-0.177979\pi$
$757$	35312.0	1.69543	0.847713	+	0.530455i	$0.177979\pi$
$758$	0	0
$759$	4956.38	0.237029
$760$	0	0
$761$	−28072.0	−1.33720	−0.668600	−	0.743622i	$-0.733106\pi$
$761$	−28072.0	−1.33720	−0.668600	+	0.743622i	$0.733106\pi$
$762$	0	0
$763$	14887.4	0.706368
$764$	0	0
$765$	18217.7	0.860996
$766$	0	0
$767$	−23305.5	−1.09715
$768$	0	0
$769$	37676.6	1.76678	0.883388	−	0.468641i	$-0.155256\pi$
$769$	37676.6	1.76678	0.883388	+	0.468641i	$0.155256\pi$
$770$	0	0
$771$	−24091.4	−1.12533
$772$	0	0
$773$	−21924.0	−1.02012	−0.510058	−	0.860140i	$-0.670376\pi$
$773$	−21924.0	−1.02012	−0.510058	+	0.860140i	$0.670376\pi$
$774$	0	0
$775$	54380.3	2.52052
$776$	0	0
$777$	−4931.51	−0.227692
$778$	0	0
$779$	1067.83	0.0491128
$780$	0	0
$781$	15244.0	0.698428
$782$	0	0
$783$	−4179.08	−0.190738
$784$	0	0
$785$	42734.0	1.94298
$786$	0	0
$787$	31223.2	1.41421	0.707107	−	0.707106i	$-0.250000\pi$
$787$	31223.2	1.41421	0.707107	+	0.707106i	$0.250000\pi$
$788$	0	0
$789$	−8621.53	−0.389017
$790$	0	0
$791$	3797.26	0.170689
$792$	0	0
$793$	30285.3	1.35620
$794$	0	0
$795$	−21123.8	−0.942370
$796$	0	0
$797$	41715.4	1.85400	0.926998	−	0.375067i	$-0.122380\pi$
$797$	41715.4	1.85400	0.926998	+	0.375067i	$0.122380\pi$
$798$	0	0
$799$	14855.4	0.657757
$800$	0	0
$801$	−11008.3	−0.485591
$802$	0	0
$803$	2380.50	0.104615
$804$	0	0
$805$	−14563.1	−0.637619
$806$	0	0
$807$	−25134.1	−1.09636
$808$	0	0
$809$	−32949.3	−1.43193	−0.715967	−	0.698134i	$-0.754014\pi$
$809$	−32949.3	−1.43193	−0.715967	+	0.698134i	$0.754014\pi$
$810$	0	0
$811$	−30285.6	−1.31131	−0.655655	−	0.755061i	$-0.727607\pi$
$811$	−30285.6	−1.31131	−0.655655	+	0.755061i	$0.727607\pi$
$812$	0	0
$813$	−2097.03	−0.0904627
$814$	0	0
$815$	6240.26	0.268205
$816$	0	0
$817$	25638.9	1.09791
$818$	0	0
$819$	5605.09	0.239143
$820$	0	0
$821$	1354.08	0.0575612	0.0287806	−	0.999586i	$-0.490838\pi$
$821$	1354.08	0.0575612	0.0287806	+	0.999586i	$0.490838\pi$
$822$	0	0
$823$	19487.5	0.825384	0.412692	−	0.910871i	$-0.364589\pi$
$823$	19487.5	0.825384	0.412692	+	0.910871i	$0.364589\pi$
$824$	0	0
$825$	−13095.6	−0.552643
$826$	0	0
$827$	34127.7	1.43499	0.717495	−	0.696563i	$-0.245289\pi$
$827$	34127.7	1.43499	0.717495	+	0.696563i	$0.245289\pi$
$828$	0	0
$829$	−31388.5	−1.31504	−0.657520	−	0.753437i	$-0.728395\pi$
$829$	−31388.5	−1.31504	−0.657520	+	0.753437i	$0.728395\pi$
$830$	0	0
$831$	11770.8	0.491366
$832$	0	0
$833$	16985.2	0.706484
$834$	0	0
$835$	78954.4	3.27225
$836$	0	0
$837$	−31289.3	−1.29213
$838$	0	0
$839$	−30992.9	−1.27532	−0.637661	−	0.770317i	$-0.720098\pi$
$839$	−30992.9	−1.27532	−0.637661	+	0.770317i	$0.720098\pi$
$840$	0	0
$841$	841.000	0.0344828
$842$	0	0
$843$	−15237.4	−0.622542
$844$	0	0
$845$	−2298.16	−0.0935610
$846$	0	0
$847$	−8667.23	−0.351605
$848$	0	0
$849$	9726.56	0.393186
$850$	0	0
$851$	17390.5	0.700515
$852$	0	0
$853$	11988.8	0.481228	0.240614	−	0.970621i	$-0.422651\pi$
$853$	11988.8	0.481228	0.240614	+	0.970621i	$0.422651\pi$
$854$	0	0
$855$	23863.0	0.954501
$856$	0	0
$857$	−46615.4	−1.85805	−0.929027	−	0.370012i	$-0.879354\pi$
$857$	−46615.4	−1.85805	−0.929027	+	0.370012i	$0.879354\pi$
$858$	0	0
$859$	21103.2	0.838221	0.419110	−	0.907935i	$-0.362342\pi$
$859$	21103.2	0.838221	0.419110	+	0.907935i	$0.362342\pi$
$860$	0	0
$861$	−361.411	−0.0143053
$862$	0	0
$863$	−26288.3	−1.03692	−0.518461	−	0.855101i	$-0.673495\pi$
$863$	−26288.3	−1.03692	−0.518461	+	0.855101i	$0.673495\pi$
$864$	0	0
$865$	3328.70	0.130843
$866$	0	0
$867$	−4192.79	−0.164238
$868$	0	0
$869$	3452.78	0.134784
$870$	0	0
$871$	−21785.1	−0.847487
$872$	0	0
$873$	12448.9	0.482626
$874$	0	0
$875$	19274.1	0.744666
$876$	0	0
$877$	25801.8	0.993459	0.496729	−	0.867905i	$-0.334534\pi$
$877$	25801.8	0.993459	0.496729	+	0.867905i	$0.334534\pi$
$878$	0	0
$879$	−18521.4	−0.710707
$880$	0	0
$881$	14409.5	0.551042	0.275521	−	0.961295i	$-0.411150\pi$
$881$	14409.5	0.551042	0.275521	+	0.961295i	$0.411150\pi$
$882$	0	0
$883$	40391.4	1.53939	0.769694	−	0.638413i	$-0.220409\pi$
$883$	40391.4	1.53939	0.769694	+	0.638413i	$0.220409\pi$
$884$	0	0
$885$	33581.6	1.27552
$886$	0	0
$887$	18395.9	0.696362	0.348181	−	0.937427i	$-0.386799\pi$
$887$	18395.9	0.696362	0.348181	+	0.937427i	$0.386799\pi$
$888$	0	0
$889$	−9340.55	−0.352387
$890$	0	0
$891$	1077.74	0.0405226
$892$	0	0
$893$	19458.9	0.729189
$894$	0	0
$895$	−50626.0	−1.89077
$896$	0	0
$897$	14650.8	0.545346
$898$	0	0
$899$	6296.67	0.233599
$900$	0	0
$901$	19497.1	0.720911
$902$	0	0
$903$	−8677.58	−0.319792
$904$	0	0
$905$	21005.7	0.771549
$906$	0	0
$907$	−15677.3	−0.573930	−0.286965	−	0.957941i	$-0.592646\pi$
$907$	−15677.3	−0.573930	−0.286965	+	0.957941i	$0.592646\pi$
$908$	0	0
$909$	10795.2	0.393898
$910$	0	0
$911$	22173.0	0.806392	0.403196	−	0.915114i	$-0.367899\pi$
$911$	22173.0	0.806392	0.403196	+	0.915114i	$0.367899\pi$
$912$	0	0
$913$	−18112.8	−0.656568
$914$	0	0
$915$	−43639.0	−1.57668
$916$	0	0
$917$	10810.5	0.389305
$918$	0	0
$919$	24630.4	0.884093	0.442047	−	0.896992i	$-0.354253\pi$
$919$	24630.4	0.884093	0.442047	+	0.896992i	$0.354253\pi$
$920$	0	0
$921$	20885.7	0.747237
$922$	0	0
$923$	45060.3	1.60691
$924$	0	0
$925$	−45948.6	−1.63328
$926$	0	0
$927$	−3496.96	−0.123900
$928$	0	0
$929$	12114.5	0.427840	0.213920	−	0.976851i	$-0.431377\pi$
$929$	12114.5	0.427840	0.213920	+	0.976851i	$0.431377\pi$
$930$	0	0
$931$	22248.5	0.783208
$932$	0	0
$933$	−5684.00	−0.199449
$934$	0	0
$935$	18119.7	0.633772
$936$	0	0
$937$	−44795.9	−1.56181	−0.780906	−	0.624649i	$-0.785242\pi$
$937$	−44795.9	−1.56181	−0.780906	+	0.624649i	$0.785242\pi$
$938$	0	0
$939$	8141.05	0.282932
$940$	0	0
$941$	40211.6	1.39305	0.696525	−	0.717532i	$-0.254728\pi$
$941$	40211.6	1.39305	0.696525	+	0.717532i	$0.254728\pi$
$942$	0	0
$943$	1274.48	0.0440115
$944$	0	0
$945$	−22139.6	−0.762116
$946$	0	0
$947$	9977.38	0.342367	0.171183	−	0.985239i	$-0.445241\pi$
$947$	9977.38	0.342367	0.171183	+	0.985239i	$0.445241\pi$
$948$	0	0
$949$	7036.61	0.240693
$950$	0	0
$951$	−19303.5	−0.658210
$952$	0	0
$953$	5358.75	0.182148	0.0910740	−	0.995844i	$-0.470970\pi$
$953$	5358.75	0.182148	0.0910740	+	0.995844i	$0.470970\pi$
$954$	0	0
$955$	48428.9	1.64097
$956$	0	0
$957$	−1516.33	−0.0512184
$958$	0	0
$959$	−5621.52	−0.189289
$960$	0	0
$961$	17352.9	0.582488
$962$	0	0
$963$	11704.3	0.391657
$964$	0	0
$965$	−23056.2	−0.769125
$966$	0	0
$967$	−34177.6	−1.13658	−0.568292	−	0.822827i	$-0.692395\pi$
$967$	−34177.6	−1.13658	−0.568292	+	0.822827i	$0.692395\pi$
$968$	0	0
$969$	16325.5	0.541230
$970$	0	0
$971$	−34564.3	−1.14235	−0.571174	−	0.820829i	$-0.693512\pi$
$971$	−34564.3	−1.14235	−0.571174	+	0.820829i	$0.693512\pi$
$972$	0	0
$973$	16560.7	0.545645
$974$	0	0
$975$	−38709.8	−1.27149
$976$	0	0
$977$	26368.2	0.863451	0.431726	−	0.902005i	$-0.357905\pi$
$977$	26368.2	0.863451	0.431726	+	0.902005i	$0.357905\pi$
$978$	0	0
$979$	−10949.0	−0.357439
$980$	0	0
$981$	−29115.3	−0.947584
$982$	0	0
$983$	−34206.8	−1.10989	−0.554947	−	0.831885i	$-0.687262\pi$
$983$	−34206.8	−1.10989	−0.554947	+	0.831885i	$0.687262\pi$
$984$	0	0
$985$	59061.2	1.91050
$986$	0	0
$987$	−6585.94	−0.212394
$988$	0	0
$989$	30600.7	0.983868
$990$	0	0
$991$	22991.7	0.736989	0.368494	−	0.929630i	$-0.379873\pi$
$991$	22991.7	0.736989	0.368494	+	0.929630i	$0.379873\pi$
$992$	0	0
$993$	−11792.7	−0.376870
$994$	0	0
$995$	−24859.0	−0.792043
$996$	0	0
$997$	18342.9	0.582674	0.291337	−	0.956620i	$-0.405900\pi$
$997$	18342.9	0.582674	0.291337	+	0.956620i	$0.405900\pi$
$998$	0	0
$999$	26437.8	0.837294

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1856.4.a.bg.1.6		9
4.3	odd	2		1856.4.a.bf.1.4		9
8.3	odd	2		928.4.a.e.1.6	yes	9
8.5	even	2		928.4.a.d.1.4	✓	9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
928.4.a.d.1.4	✓	9	8.5	even	2
928.4.a.e.1.6	yes	9	8.3	odd	2
1856.4.a.bf.1.4		9	4.3	odd	2
1856.4.a.bg.1.6		9	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

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

\(f(q)\)	\(=\)	\(q+3.39022 q^{3} +19.3767 q^{5} +7.92881 q^{7} -15.5064 q^{9} +O(q^{10})\)	\(q+3.39022 q^{3} +19.3767 q^{5} +7.92881 q^{7} -15.5064 q^{9} -15.4230 q^{11} -45.5894 q^{13} +65.6911 q^{15} -60.6323 q^{17} -79.4211 q^{19} +26.8804 q^{21} -94.7913 q^{23} +250.455 q^{25} -144.106 q^{27} +29.0000 q^{29} +217.126 q^{31} -52.2873 q^{33} +153.634 q^{35} -183.461 q^{37} -154.558 q^{39} -13.4451 q^{41} -322.822 q^{43} -300.462 q^{45} -245.009 q^{47} -280.134 q^{49} -205.557 q^{51} -321.562 q^{53} -298.846 q^{55} -269.255 q^{57} +511.205 q^{59} -664.306 q^{61} -122.947 q^{63} -883.370 q^{65} +477.855 q^{67} -321.363 q^{69} -988.394 q^{71} -154.347 q^{73} +849.097 q^{75} -122.286 q^{77} -223.873 q^{79} -69.8789 q^{81} +1174.40 q^{83} -1174.85 q^{85} +98.3164 q^{87} +709.918 q^{89} -361.470 q^{91} +736.107 q^{93} -1538.91 q^{95} -802.826 q^{97} +239.155 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q + 4 q^{3} - 10 q^{5} - 12 q^{7} + 49 q^{9}+O(q^{10}) \) 9 * q + 4 * q^3 - 10 * q^5 - 12 * q^7 + 49 * q^9	\( 9 q + 4 q^{3} - 10 q^{5} - 12 q^{7} + 49 q^{9} + 64 q^{11} - 70 q^{13} - 170 q^{15} - 66 q^{17} + 42 q^{19} - 76 q^{21} - 40 q^{23} + 111 q^{25} + 322 q^{27} + 261 q^{29} + 64 q^{31} - 52 q^{33} + 496 q^{35} + 54 q^{37} - 590 q^{39} - 378 q^{41} - 32 q^{43} - 1046 q^{45} - 1164 q^{47} - 351 q^{49} + 376 q^{51} - 278 q^{53} - 614 q^{55} + 28 q^{57} + 640 q^{59} - 1054 q^{61} - 1660 q^{63} - 708 q^{65} + 1184 q^{67} - 188 q^{69} - 1988 q^{71} - 750 q^{73} + 3126 q^{75} - 1260 q^{77} - 2916 q^{79} + 293 q^{81} + 2832 q^{83} - 56 q^{85} + 116 q^{87} - 370 q^{89} + 3016 q^{91} + 1696 q^{93} - 4412 q^{95} - 2234 q^{97} + 4118 q^{99}+O(q^{100}) \) 9 * q + 4 * q^3 - 10 * q^5 - 12 * q^7 + 49 * q^9 + 64 * q^11 - 70 * q^13 - 170 * q^15 - 66 * q^17 + 42 * q^19 - 76 * q^21 - 40 * q^23 + 111 * q^25 + 322 * q^27 + 261 * q^29 + 64 * q^31 - 52 * q^33 + 496 * q^35 + 54 * q^37 - 590 * q^39 - 378 * q^41 - 32 * q^43 - 1046 * q^45 - 1164 * q^47 - 351 * q^49 + 376 * q^51 - 278 * q^53 - 614 * q^55 + 28 * q^57 + 640 * q^59 - 1054 * q^61 - 1660 * q^63 - 708 * q^65 + 1184 * q^67 - 188 * q^69 - 1988 * q^71 - 750 * q^73 + 3126 * q^75 - 1260 * q^77 - 2916 * q^79 + 293 * q^81 + 2832 * q^83 - 56 * q^85 + 116 * q^87 - 370 * q^89 + 3016 * q^91 + 1696 * q^93 - 4412 * q^95 - 2234 * q^97 + 4118 * q^99

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