LMFDB - Embedded newform 232.4.a.b.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [232,4,Mod(1,232)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(232, 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("232.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 232 = 2^{3} \cdot 29 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	232.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:	$13.6884431213$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.229.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 4x - 1 $ x^3 - 4*x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-0.254102$ of defining polynomial
Character	$\chi$	$=$	232.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+9.04399 q^{3} +2.15672 q^{5} +27.7969 q^{7} +54.7938 q^{9} +O(q^{10})$	$q+9.04399 q^{3} +2.15672 q^{5} +27.7969 q^{7} +54.7938 q^{9} -0.351483 q^{11} -65.3652 q^{13} +19.5054 q^{15} -68.3296 q^{17} +89.8292 q^{19} +251.395 q^{21} +110.709 q^{23} -120.349 q^{25} +251.367 q^{27} -29.0000 q^{29} -258.256 q^{31} -3.17881 q^{33} +59.9501 q^{35} -128.225 q^{37} -591.163 q^{39} +283.005 q^{41} +339.318 q^{43} +118.175 q^{45} -147.455 q^{47} +429.668 q^{49} -617.972 q^{51} +518.354 q^{53} -0.758050 q^{55} +812.414 q^{57} -102.347 q^{59} -791.964 q^{61} +1523.10 q^{63} -140.975 q^{65} +287.326 q^{67} +1001.25 q^{69} -546.837 q^{71} -260.144 q^{73} -1088.43 q^{75} -9.77013 q^{77} -204.498 q^{79} +793.927 q^{81} -949.003 q^{83} -147.368 q^{85} -262.276 q^{87} +1390.02 q^{89} -1816.95 q^{91} -2335.67 q^{93} +193.736 q^{95} +1228.30 q^{97} -19.2591 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 6 q^{3} + 4 q^{5} + 16 q^{7} + 45 q^{9}+O(q^{10}) $ 3 * q + 6 * q^3 + 4 * q^5 + 16 * q^7 + 45 * q^9	$ 3 q + 6 q^{3} + 4 q^{5} + 16 q^{7} + 45 q^{9} - 2 q^{11} + 28 q^{13} + 136 q^{15} - 66 q^{17} - 66 q^{19} + 472 q^{21} + 176 q^{23} - 9 q^{25} + 228 q^{27} - 87 q^{29} - 190 q^{31} + 154 q^{33} + 660 q^{35} + 442 q^{37} - 656 q^{39} + 1162 q^{41} + 30 q^{43} - 254 q^{45} - 738 q^{47} + 851 q^{49} - 576 q^{51} + 312 q^{53} + 464 q^{55} + 684 q^{57} + 44 q^{59} + 54 q^{61} + 964 q^{63} + 178 q^{65} - 116 q^{67} + 812 q^{69} - 1200 q^{71} - 1118 q^{73} - 1038 q^{75} + 792 q^{77} - 2262 q^{79} + 15 q^{81} - 1804 q^{83} - 8 q^{85} - 174 q^{87} + 1578 q^{89} - 1972 q^{91} - 706 q^{93} - 1052 q^{95} + 1450 q^{97} - 482 q^{99}+O(q^{100}) $ 3 * q + 6 * q^3 + 4 * q^5 + 16 * q^7 + 45 * q^9 - 2 * q^11 + 28 * q^13 + 136 * q^15 - 66 * q^17 - 66 * q^19 + 472 * q^21 + 176 * q^23 - 9 * q^25 + 228 * q^27 - 87 * q^29 - 190 * q^31 + 154 * q^33 + 660 * q^35 + 442 * q^37 - 656 * q^39 + 1162 * q^41 + 30 * q^43 - 254 * q^45 - 738 * q^47 + 851 * q^49 - 576 * q^51 + 312 * q^53 + 464 * q^55 + 684 * q^57 + 44 * q^59 + 54 * q^61 + 964 * q^63 + 178 * q^65 - 116 * q^67 + 812 * q^69 - 1200 * q^71 - 1118 * q^73 - 1038 * q^75 + 792 * q^77 - 2262 * q^79 + 15 * q^81 - 1804 * q^83 - 8 * q^85 - 174 * q^87 + 1578 * q^89 - 1972 * q^91 - 706 * q^93 - 1052 * q^95 + 1450 * q^97 - 482 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	9.04399	1.74052	0.870259	−	0.492595i	$-0.163952\pi$
$3$	9.04399	1.74052	0.870259	+	0.492595i	$0.163952\pi$
$4$	0	0
$5$	2.15672	0.192903	0.0964515	−	0.995338i	$-0.469251\pi$
$5$	2.15672	0.192903	0.0964515	+	0.995338i	$0.469251\pi$
$6$	0	0
$7$	27.7969	1.50089	0.750446	−	0.660932i	$-0.229839\pi$
$7$	27.7969	1.50089	0.750446	+	0.660932i	$0.229839\pi$
$8$	0	0
$9$	54.7938	2.02940
$10$	0	0
$11$	−0.351483	−0.00963419	−0.00481709	−	0.999988i	$-0.501533\pi$
$11$	−0.351483	−0.00963419	−0.00481709	+	0.999988i	$0.501533\pi$
$12$	0	0
$13$	−65.3652	−1.39454	−0.697271	−	0.716807i	$-0.745603\pi$
$13$	−65.3652	−1.39454	−0.697271	+	0.716807i	$0.745603\pi$
$14$	0	0
$15$	19.5054	0.335751
$16$	0	0
$17$	−68.3296	−0.974845	−0.487422	−	0.873166i	$-0.662063\pi$
$17$	−68.3296	−0.974845	−0.487422	+	0.873166i	$0.662063\pi$
$18$	0	0
$19$	89.8292	1.08464	0.542322	−	0.840171i	$-0.317545\pi$
$19$	89.8292	1.08464	0.542322	+	0.840171i	$0.317545\pi$
$20$	0	0
$21$	251.395	2.61233
$22$	0	0
$23$	110.709	1.00367	0.501834	−	0.864964i	$-0.332659\pi$
$23$	110.709	1.00367	0.501834	+	0.864964i	$0.332659\pi$
$24$	0	0
$25$	−120.349	−0.962788
$26$	0	0
$27$	251.367	1.79169
$28$	0	0
$29$	−29.0000	−0.185695
$29$	−29.0000	−0.185695
$30$	0	0
$31$	−258.256	−1.49626	−0.748132	−	0.663550i	$-0.769049\pi$
$31$	−258.256	−1.49626	−0.748132	+	0.663550i	$0.769049\pi$
$32$	0	0
$33$	−3.17881	−0.0167685
$34$	0	0
$35$	59.9501	0.289526
$36$	0	0
$37$	−128.225	−0.569733	−0.284866	−	0.958567i	$-0.591949\pi$
$37$	−128.225	−0.569733	−0.284866	+	0.958567i	$0.591949\pi$
$38$	0	0
$39$	−591.163	−2.42723
$40$	0	0
$41$	283.005	1.07800	0.539000	−	0.842306i	$-0.318802\pi$
$41$	283.005	1.07800	0.539000	+	0.842306i	$0.318802\pi$
$42$	0	0
$43$	339.318	1.20338	0.601692	−	0.798728i	$-0.294493\pi$
$43$	339.318	1.20338	0.601692	+	0.798728i	$0.294493\pi$
$44$	0	0
$45$	118.175	0.391477
$46$	0	0
$47$	−147.455	−0.457628	−0.228814	−	0.973470i	$-0.573485\pi$
$47$	−147.455	−0.457628	−0.228814	+	0.973470i	$0.573485\pi$
$48$	0	0
$49$	429.668	1.25268
$50$	0	0
$51$	−617.972	−1.69673
$52$	0	0
$53$	518.354	1.34342	0.671712	−	0.740813i	$-0.265559\pi$
$53$	518.354	1.34342	0.671712	+	0.740813i	$0.265559\pi$
$54$	0	0
$55$	−0.758050	−0.00185846
$56$	0	0
$57$	812.414	1.88784
$58$	0	0
$59$	−102.347	−0.225838	−0.112919	−	0.993604i	$-0.536020\pi$
$59$	−102.347	−0.225838	−0.112919	+	0.993604i	$0.536020\pi$
$60$	0	0
$61$	−791.964	−1.66230	−0.831152	−	0.556045i	$-0.812318\pi$
$61$	−791.964	−1.66230	−0.831152	+	0.556045i	$0.812318\pi$
$62$	0	0
$63$	1523.10	3.04591
$64$	0	0
$65$	−140.975	−0.269011
$66$	0	0
$67$	287.326	0.523918	0.261959	−	0.965079i	$-0.415632\pi$
$67$	287.326	0.523918	0.261959	+	0.965079i	$0.415632\pi$
$68$	0	0
$69$	1001.25	1.74690
$70$	0	0
$71$	−546.837	−0.914050	−0.457025	−	0.889454i	$-0.651085\pi$
$71$	−546.837	−0.914050	−0.457025	+	0.889454i	$0.651085\pi$
$72$	0	0
$73$	−260.144	−0.417089	−0.208545	−	0.978013i	$-0.566873\pi$
$73$	−260.144	−0.417089	−0.208545	+	0.978013i	$0.566873\pi$
$74$	0	0
$75$	−1088.43	−1.67575
$76$	0	0
$77$	−9.77013	−0.0144599
$78$	0	0
$79$	−204.498	−0.291238	−0.145619	−	0.989341i	$-0.546517\pi$
$79$	−204.498	−0.291238	−0.145619	+	0.989341i	$0.546517\pi$
$80$	0	0
$81$	793.927	1.08906
$82$	0	0
$83$	−949.003	−1.25502	−0.627510	−	0.778609i	$-0.715926\pi$
$83$	−949.003	−1.25502	−0.627510	+	0.778609i	$0.715926\pi$
$84$	0	0
$85$	−147.368	−0.188050
$86$	0	0
$87$	−262.276	−0.323206
$88$	0	0
$89$	1390.02	1.65553	0.827765	−	0.561074i	$-0.189612\pi$
$89$	1390.02	1.65553	0.827765	+	0.561074i	$0.189612\pi$
$90$	0	0
$91$	−1816.95	−2.09306
$92$	0	0
$93$	−2335.67	−2.60427
$94$	0	0
$95$	193.736	0.209231
$96$	0	0
$97$	1228.30	1.28572	0.642861	−	0.765983i	$-0.277747\pi$
$97$	1228.30	1.28572	0.642861	+	0.765983i	$0.277747\pi$
$98$	0	0
$99$	−19.2591	−0.0195516
$100$	0	0
$101$	1090.25	1.07410	0.537049	−	0.843551i	$-0.319539\pi$
$101$	1090.25	1.07410	0.537049	+	0.843551i	$0.319539\pi$
$102$	0	0
$103$	−1226.95	−1.17374	−0.586869	−	0.809682i	$-0.699640\pi$
$103$	−1226.95	−1.17374	−0.586869	+	0.809682i	$0.699640\pi$
$104$	0	0
$105$	542.189	0.503926
$106$	0	0
$107$	−365.324	−0.330067	−0.165034	−	0.986288i	$-0.552773\pi$
$107$	−365.324	−0.330067	−0.165034	+	0.986288i	$0.552773\pi$
$108$	0	0
$109$	−1759.55	−1.54619	−0.773093	−	0.634292i	$-0.781292\pi$
$109$	−1759.55	−1.54619	−0.773093	+	0.634292i	$0.781292\pi$
$110$	0	0
$111$	−1159.67	−0.991629
$112$	0	0
$113$	−1764.47	−1.46891	−0.734457	−	0.678655i	$-0.762563\pi$
$113$	−1764.47	−1.46891	−0.734457	+	0.678655i	$0.762563\pi$
$114$	0	0
$115$	238.768	0.193610
$116$	0	0
$117$	−3581.61	−2.83008
$118$	0	0
$119$	−1899.35	−1.46314
$120$	0	0
$121$	−1330.88	−0.999907
$122$	0	0
$123$	2559.50	1.87628
$124$	0	0
$125$	−529.148	−0.378628
$126$	0	0
$127$	−1121.29	−0.783453	−0.391726	−	0.920082i	$-0.628122\pi$
$127$	−1121.29	−0.783453	−0.391726	+	0.920082i	$0.628122\pi$
$128$	0	0
$129$	3068.79	2.09451
$130$	0	0
$131$	1276.32	0.851240	0.425620	−	0.904902i	$-0.360056\pi$
$131$	1276.32	0.851240	0.425620	+	0.904902i	$0.360056\pi$
$132$	0	0
$133$	2496.97	1.62793
$134$	0	0
$135$	542.128	0.345622
$136$	0	0
$137$	−1564.49	−0.975644	−0.487822	−	0.872943i	$-0.662208\pi$
$137$	−1564.49	−0.975644	−0.487822	+	0.872943i	$0.662208\pi$
$138$	0	0
$139$	2231.83	1.36188	0.680940	−	0.732339i	$-0.261572\pi$
$139$	2231.83	1.36188	0.680940	+	0.732339i	$0.261572\pi$
$140$	0	0
$141$	−1333.58	−0.796509
$142$	0	0
$143$	22.9748	0.0134353
$144$	0	0
$145$	−62.5449	−0.0358212
$146$	0	0
$147$	3885.91	2.18030
$148$	0	0
$149$	−3474.95	−1.91060	−0.955300	−	0.295639i	$-0.904467\pi$
$149$	−3474.95	−1.91060	−0.955300	+	0.295639i	$0.904467\pi$
$150$	0	0
$151$	1061.60	0.572131	0.286066	−	0.958210i	$-0.407652\pi$
$151$	1061.60	0.572131	0.286066	+	0.958210i	$0.407652\pi$
$152$	0	0
$153$	−3744.04	−1.97835
$154$	0	0
$155$	−556.986	−0.288634
$156$	0	0
$157$	−1664.40	−0.846073	−0.423036	−	0.906113i	$-0.639036\pi$
$157$	−1664.40	−0.846073	−0.423036	+	0.906113i	$0.639036\pi$
$158$	0	0
$159$	4687.99	2.33825
$160$	0	0
$161$	3077.36	1.50640
$162$	0	0
$163$	−239.164	−0.114925	−0.0574624	−	0.998348i	$-0.518301\pi$
$163$	−239.164	−0.114925	−0.0574624	+	0.998348i	$0.518301\pi$
$164$	0	0
$165$	−6.85580	−0.00323469
$166$	0	0
$167$	1330.43	0.616477	0.308238	−	0.951309i	$-0.400261\pi$
$167$	1330.43	0.616477	0.308238	+	0.951309i	$0.400261\pi$
$168$	0	0
$169$	2075.61	0.944749
$170$	0	0
$171$	4922.08	2.20118
$172$	0	0
$173$	3007.36	1.32165	0.660824	−	0.750541i	$-0.270207\pi$
$173$	3007.36	1.32165	0.660824	+	0.750541i	$0.270207\pi$
$174$	0	0
$175$	−3345.32	−1.44504
$176$	0	0
$177$	−925.627	−0.393076
$178$	0	0
$179$	722.292	0.301601	0.150801	−	0.988564i	$-0.451815\pi$
$179$	722.292	0.301601	0.150801	+	0.988564i	$0.451815\pi$
$180$	0	0
$181$	2763.91	1.13503	0.567514	−	0.823364i	$-0.307905\pi$
$181$	2763.91	1.13503	0.567514	+	0.823364i	$0.307905\pi$
$182$	0	0
$183$	−7162.51	−2.89327
$184$	0	0
$185$	−276.546	−0.109903
$186$	0	0
$187$	24.0167	0.00939184
$188$	0	0
$189$	6987.22	2.68913
$190$	0	0
$191$	719.960	0.272746	0.136373	−	0.990658i	$-0.456455\pi$
$191$	719.960	0.272746	0.136373	+	0.990658i	$0.456455\pi$
$192$	0	0
$193$	1020.08	0.380452	0.190226	−	0.981740i	$-0.439078\pi$
$193$	1020.08	0.380452	0.190226	+	0.981740i	$0.439078\pi$
$194$	0	0
$195$	−1274.97	−0.468219
$196$	0	0
$197$	182.348	0.0659482	0.0329741	−	0.999456i	$-0.489502\pi$
$197$	182.348	0.0659482	0.0329741	+	0.999456i	$0.489502\pi$
$198$	0	0
$199$	−3150.60	−1.12231	−0.561156	−	0.827710i	$-0.689643\pi$
$199$	−3150.60	−1.12231	−0.561156	+	0.827710i	$0.689643\pi$
$200$	0	0
$201$	2598.58	0.911888
$202$	0	0
$203$	−806.110	−0.278709
$204$	0	0
$205$	610.363	0.207949
$206$	0	0
$207$	6066.15	2.03684
$208$	0	0
$209$	−31.5734	−0.0104497
$210$	0	0
$211$	2028.36	0.661793	0.330897	−	0.943667i	$-0.392649\pi$
$211$	2028.36	0.661793	0.330897	+	0.943667i	$0.392649\pi$
$212$	0	0
$213$	−4945.59	−1.59092
$214$	0	0
$215$	731.815	0.232136
$216$	0	0
$217$	−7178.72	−2.24573
$218$	0	0
$219$	−2352.74	−0.725951
$220$	0	0
$221$	4466.38	1.35946
$222$	0	0
$223$	−1942.22	−0.583232	−0.291616	−	0.956535i	$-0.594193\pi$
$223$	−1942.22	−0.583232	−0.291616	+	0.956535i	$0.594193\pi$
$224$	0	0
$225$	−6594.35	−1.95388
$226$	0	0
$227$	3476.41	1.01646	0.508232	−	0.861220i	$-0.330299\pi$
$227$	3476.41	1.01646	0.508232	+	0.861220i	$0.330299\pi$
$228$	0	0
$229$	3015.30	0.870116	0.435058	−	0.900402i	$-0.356728\pi$
$229$	3015.30	0.870116	0.435058	+	0.900402i	$0.356728\pi$
$230$	0	0
$231$	−88.3610	−0.0251677
$232$	0	0
$233$	5510.20	1.54929	0.774646	−	0.632395i	$-0.217928\pi$
$233$	5510.20	1.54929	0.774646	+	0.632395i	$0.217928\pi$
$234$	0	0
$235$	−318.019	−0.0882778
$236$	0	0
$237$	−1849.48	−0.506905
$238$	0	0
$239$	1742.92	0.471716	0.235858	−	0.971787i	$-0.424210\pi$
$239$	1742.92	0.471716	0.235858	+	0.971787i	$0.424210\pi$
$240$	0	0
$241$	−1188.65	−0.317707	−0.158854	−	0.987302i	$-0.550780\pi$
$241$	−1188.65	−0.317707	−0.158854	+	0.987302i	$0.550780\pi$
$242$	0	0
$243$	393.366	0.103846
$244$	0	0
$245$	926.673	0.241645
$246$	0	0
$247$	−5871.70	−1.51258
$248$	0	0
$249$	−8582.78	−2.18438
$250$	0	0
$251$	4679.47	1.17675	0.588377	−	0.808587i	$-0.299767\pi$
$251$	4679.47	1.17675	0.588377	+	0.808587i	$0.299767\pi$
$252$	0	0
$253$	−38.9122	−0.00966952
$254$	0	0
$255$	−1332.79	−0.327305
$256$	0	0
$257$	2223.42	0.539663	0.269831	−	0.962908i	$-0.413032\pi$
$257$	2223.42	0.539663	0.269831	+	0.962908i	$0.413032\pi$
$258$	0	0
$259$	−3564.27	−0.855107
$260$	0	0
$261$	−1589.02	−0.376850
$262$	0	0
$263$	6720.95	1.57579	0.787893	−	0.615813i	$-0.211172\pi$
$263$	6720.95	1.57579	0.787893	+	0.615813i	$0.211172\pi$
$264$	0	0
$265$	1117.95	0.259150
$266$	0	0
$267$	12571.4	2.88148
$268$	0	0
$269$	−2842.85	−0.644356	−0.322178	−	0.946679i	$-0.604415\pi$
$269$	−2842.85	−0.644356	−0.322178	+	0.946679i	$0.604415\pi$
$270$	0	0
$271$	−3793.02	−0.850219	−0.425109	−	0.905142i	$-0.639764\pi$
$271$	−3793.02	−0.850219	−0.425109	+	0.905142i	$0.639764\pi$
$272$	0	0
$273$	−16432.5	−3.64300
$274$	0	0
$275$	42.3005	0.00927569
$276$	0	0
$277$	6698.19	1.45291	0.726453	−	0.687216i	$-0.241167\pi$
$277$	6698.19	1.45291	0.726453	+	0.687216i	$0.241167\pi$
$278$	0	0
$279$	−14150.8	−3.03652
$280$	0	0
$281$	8532.95	1.81151	0.905753	−	0.423807i	$-0.139306\pi$
$281$	8532.95	1.81151	0.905753	+	0.423807i	$0.139306\pi$
$282$	0	0
$283$	−9100.83	−1.91162	−0.955810	−	0.293986i	$-0.905018\pi$
$283$	−9100.83	−1.91162	−0.955810	+	0.293986i	$0.905018\pi$
$284$	0	0
$285$	1752.15	0.364170
$286$	0	0
$287$	7866.66	1.61796
$288$	0	0
$289$	−244.069	−0.0496781
$290$	0	0
$291$	11108.7	2.23782
$292$	0	0
$293$	6416.32	1.27934	0.639668	−	0.768651i	$-0.279072\pi$
$293$	6416.32	1.27934	0.639668	+	0.768651i	$0.279072\pi$
$294$	0	0
$295$	−220.734	−0.0435649
$296$	0	0
$297$	−88.3511	−0.0172615
$298$	0	0
$299$	−7236.50	−1.39966
$300$	0	0
$301$	9431.99	1.80615
$302$	0	0
$303$	9860.20	1.86948
$304$	0	0
$305$	−1708.04	−0.320663
$306$	0	0
$307$	9836.12	1.82859	0.914295	−	0.405050i	$-0.132746\pi$
$307$	9836.12	1.82859	0.914295	+	0.405050i	$0.132746\pi$
$308$	0	0
$309$	−11096.5	−2.04291
$310$	0	0
$311$	6806.95	1.24112	0.620558	−	0.784161i	$-0.286906\pi$
$311$	6806.95	1.24112	0.620558	+	0.784161i	$0.286906\pi$
$312$	0	0
$313$	1860.04	0.335896	0.167948	−	0.985796i	$-0.446286\pi$
$313$	1860.04	0.335896	0.167948	+	0.985796i	$0.446286\pi$
$314$	0	0
$315$	3284.90	0.587565
$316$	0	0
$317$	2326.51	0.412208	0.206104	−	0.978530i	$-0.433922\pi$
$317$	2326.51	0.412208	0.206104	+	0.978530i	$0.433922\pi$
$318$	0	0
$319$	10.1930	0.00178902
$320$	0	0
$321$	−3303.98	−0.574487
$322$	0	0
$323$	−6137.99	−1.05736
$324$	0	0
$325$	7866.61	1.34265
$326$	0	0
$327$	−15913.4	−2.69116
$328$	0	0
$329$	−4098.79	−0.686850
$330$	0	0
$331$	−1668.98	−0.277146	−0.138573	−	0.990352i	$-0.544252\pi$
$331$	−1668.98	−0.277146	−0.138573	+	0.990352i	$0.544252\pi$
$332$	0	0
$333$	−7025.95	−1.15622
$334$	0	0
$335$	619.682	0.101065
$336$	0	0
$337$	−9794.95	−1.58328	−0.791639	−	0.610989i	$-0.790772\pi$
$337$	−9794.95	−1.58328	−0.791639	+	0.610989i	$0.790772\pi$
$338$	0	0
$339$	−15957.8	−2.55667
$340$	0	0
$341$	90.7726	0.0144153
$342$	0	0
$343$	2409.09	0.379238
$344$	0	0
$345$	2159.41	0.336982
$346$	0	0
$347$	−2296.52	−0.355285	−0.177642	−	0.984095i	$-0.556847\pi$
$347$	−2296.52	−0.355285	−0.177642	+	0.984095i	$0.556847\pi$
$348$	0	0
$349$	9856.35	1.51174	0.755871	−	0.654720i	$-0.227214\pi$
$349$	9856.35	1.51174	0.755871	+	0.654720i	$0.227214\pi$
$350$	0	0
$351$	−16430.7	−2.49858
$352$	0	0
$353$	−10005.5	−1.50860	−0.754301	−	0.656528i	$-0.772024\pi$
$353$	−10005.5	−1.50860	−0.754301	+	0.656528i	$0.772024\pi$
$354$	0	0
$355$	−1179.37	−0.176323
$356$	0	0
$357$	−17177.7	−2.54661
$358$	0	0
$359$	176.590	0.0259612	0.0129806	−	0.999916i	$-0.495868\pi$
$359$	176.590	0.0259612	0.0129806	+	0.999916i	$0.495868\pi$
$360$	0	0
$361$	1210.28	0.176451
$362$	0	0
$363$	−12036.4	−1.74036
$364$	0	0
$365$	−561.057	−0.0804577
$366$	0	0
$367$	50.2297	0.00714433	0.00357216	−	0.999994i	$-0.498863\pi$
$367$	50.2297	0.00714433	0.00357216	+	0.999994i	$0.498863\pi$
$368$	0	0
$369$	15506.9	2.18769
$370$	0	0
$371$	14408.6	2.01633
$372$	0	0
$373$	−10666.6	−1.48068	−0.740340	−	0.672232i	$-0.765336\pi$
$373$	−10666.6	−1.48068	−0.740340	+	0.672232i	$0.765336\pi$
$374$	0	0
$375$	−4785.61	−0.659008
$376$	0	0
$377$	1895.59	0.258960
$378$	0	0
$379$	−4158.76	−0.563644	−0.281822	−	0.959467i	$-0.590939\pi$
$379$	−4158.76	−0.563644	−0.281822	+	0.959467i	$0.590939\pi$
$380$	0	0
$381$	−10140.9	−1.36361
$382$	0	0
$383$	−10572.2	−1.41049	−0.705243	−	0.708965i	$-0.749162\pi$
$383$	−10572.2	−1.41049	−0.705243	+	0.708965i	$0.749162\pi$
$384$	0	0
$385$	−21.0714	−0.00278935
$386$	0	0
$387$	18592.5	2.44215
$388$	0	0
$389$	−8211.23	−1.07025	−0.535123	−	0.844774i	$-0.679735\pi$
$389$	−8211.23	−1.07025	−0.535123	+	0.844774i	$0.679735\pi$
$390$	0	0
$391$	−7564.68	−0.978420
$392$	0	0
$393$	11543.0	1.48160
$394$	0	0
$395$	−441.045	−0.0561807
$396$	0	0
$397$	4879.38	0.616849	0.308424	−	0.951249i	$-0.400198\pi$
$397$	4879.38	0.616849	0.308424	+	0.951249i	$0.400198\pi$
$398$	0	0
$399$	22582.6	2.83344
$400$	0	0
$401$	7570.58	0.942785	0.471393	−	0.881923i	$-0.343751\pi$
$401$	7570.58	0.942785	0.471393	+	0.881923i	$0.343751\pi$
$402$	0	0
$403$	16881.0	2.08660
$404$	0	0
$405$	1712.28	0.210084
$406$	0	0
$407$	45.0690	0.00548891
$408$	0	0
$409$	−11758.1	−1.42152	−0.710760	−	0.703435i	$-0.751649\pi$
$409$	−11758.1	−1.42152	−0.710760	+	0.703435i	$0.751649\pi$
$410$	0	0
$411$	−14149.2	−1.69812
$412$	0	0
$413$	−2844.93	−0.338959
$414$	0	0
$415$	−2046.73	−0.242097
$416$	0	0
$417$	20184.6	2.37037
$418$	0	0
$419$	10150.6	1.18351	0.591755	−	0.806118i	$-0.298435\pi$
$419$	10150.6	1.18351	0.591755	+	0.806118i	$0.298435\pi$
$420$	0	0
$421$	2822.78	0.326779	0.163390	−	0.986562i	$-0.447757\pi$
$421$	2822.78	0.326779	0.163390	+	0.986562i	$0.447757\pi$
$422$	0	0
$423$	−8079.62	−0.928710
$424$	0	0
$425$	8223.37	0.938569
$426$	0	0
$427$	−22014.1	−2.49494
$428$	0	0
$429$	207.784	0.0233843
$430$	0	0
$431$	−6663.84	−0.744747	−0.372373	−	0.928083i	$-0.621456\pi$
$431$	−6663.84	−0.744747	−0.372373	+	0.928083i	$0.621456\pi$
$432$	0	0
$433$	−6591.27	−0.731539	−0.365769	−	0.930706i	$-0.619194\pi$
$433$	−6591.27	−0.731539	−0.365769	+	0.930706i	$0.619194\pi$
$434$	0	0
$435$	−565.656	−0.0623474
$436$	0	0
$437$	9944.87	1.08862
$438$	0	0
$439$	12121.1	1.31778	0.658891	−	0.752238i	$-0.271026\pi$
$439$	12121.1	1.31778	0.658891	+	0.752238i	$0.271026\pi$
$440$	0	0
$441$	23543.1	2.54218
$442$	0	0
$443$	9017.80	0.967152	0.483576	−	0.875302i	$-0.339338\pi$
$443$	9017.80	0.967152	0.483576	+	0.875302i	$0.339338\pi$
$444$	0	0
$445$	2997.89	0.319357
$446$	0	0
$447$	−31427.5	−3.32543
$448$	0	0
$449$	−4485.59	−0.471466	−0.235733	−	0.971818i	$-0.575749\pi$
$449$	−4485.59	−0.471466	−0.235733	+	0.971818i	$0.575749\pi$
$450$	0	0
$451$	−99.4714	−0.0103856
$452$	0	0
$453$	9601.11	0.995805
$454$	0	0
$455$	−3918.66	−0.403757
$456$	0	0
$457$	7305.52	0.747785	0.373893	−	0.927472i	$-0.378023\pi$
$457$	7305.52	0.747785	0.373893	+	0.927472i	$0.378023\pi$
$458$	0	0
$459$	−17175.8	−1.74662
$460$	0	0
$461$	1174.43	0.118652	0.0593259	−	0.998239i	$-0.481105\pi$
$461$	1174.43	0.118652	0.0593259	+	0.998239i	$0.481105\pi$
$462$	0	0
$463$	7866.43	0.789598	0.394799	−	0.918768i	$-0.370814\pi$
$463$	7866.43	0.789598	0.394799	+	0.918768i	$0.370814\pi$
$464$	0	0
$465$	−5037.38	−0.502372
$466$	0	0
$467$	12691.8	1.25762	0.628810	−	0.777559i	$-0.283542\pi$
$467$	12691.8	1.25762	0.628810	+	0.777559i	$0.283542\pi$
$468$	0	0
$469$	7986.78	0.786344
$470$	0	0
$471$	−15052.8	−1.47260
$472$	0	0
$473$	−119.265	−0.0115936
$474$	0	0
$475$	−10810.8	−1.04428
$476$	0	0
$477$	28402.6	2.72634
$478$	0	0
$479$	11226.1	1.07084	0.535420	−	0.844586i	$-0.320153\pi$
$479$	11226.1	1.07084	0.535420	+	0.844586i	$0.320153\pi$
$480$	0	0
$481$	8381.48	0.794516
$482$	0	0
$483$	27831.6	2.62191
$484$	0	0
$485$	2649.10	0.248020
$486$	0	0
$487$	2493.14	0.231981	0.115991	−	0.993250i	$-0.462996\pi$
$487$	2493.14	0.231981	0.115991	+	0.993250i	$0.462996\pi$
$488$	0	0
$489$	−2163.00	−0.200029
$490$	0	0
$491$	11346.8	1.04292	0.521458	−	0.853277i	$-0.325388\pi$
$491$	11346.8	1.04292	0.521458	+	0.853277i	$0.325388\pi$
$492$	0	0
$493$	1981.56	0.181024
$494$	0	0
$495$	−41.5364	−0.00377156
$496$	0	0
$497$	−15200.4	−1.37189
$498$	0	0
$499$	−19216.1	−1.72391	−0.861956	−	0.506982i	$-0.830761\pi$
$499$	−19216.1	−1.72391	−0.861956	+	0.506982i	$0.830761\pi$
$500$	0	0
$501$	12032.4	1.07299
$502$	0	0
$503$	−9284.26	−0.822991	−0.411496	−	0.911412i	$-0.634993\pi$
$503$	−9284.26	−0.822991	−0.411496	+	0.911412i	$0.634993\pi$
$504$	0	0
$505$	2351.36	0.207196
$506$	0	0
$507$	18771.8	1.64435
$508$	0	0
$509$	2987.03	0.260113	0.130057	−	0.991507i	$-0.458484\pi$
$509$	2987.03	0.260113	0.130057	+	0.991507i	$0.458484\pi$
$510$	0	0
$511$	−7231.19	−0.626006
$512$	0	0
$513$	22580.1	1.94334
$514$	0	0
$515$	−2646.19	−0.226418
$516$	0	0
$517$	51.8279	0.00440887
$518$	0	0
$519$	27198.5	2.30035
$520$	0	0
$521$	11082.9	0.931962	0.465981	−	0.884795i	$-0.345702\pi$
$521$	11082.9	0.931962	0.465981	+	0.884795i	$0.345702\pi$
$522$	0	0
$523$	13718.6	1.14698	0.573492	−	0.819211i	$-0.305588\pi$
$523$	13718.6	1.14698	0.573492	+	0.819211i	$0.305588\pi$
$524$	0	0
$525$	−30255.0	−2.51512
$526$	0	0
$527$	17646.5	1.45862
$528$	0	0
$529$	89.4044	0.00734811
$530$	0	0
$531$	−5607.99	−0.458316
$532$	0	0
$533$	−18498.7	−1.50332
$534$	0	0
$535$	−787.901	−0.0636709
$536$	0	0
$537$	6532.40	0.524942
$538$	0	0
$539$	−151.021	−0.0120685
$540$	0	0
$541$	9946.06	0.790415	0.395207	−	0.918592i	$-0.370673\pi$
$541$	9946.06	0.790415	0.395207	+	0.918592i	$0.370673\pi$
$542$	0	0
$543$	24996.8	1.97553
$544$	0	0
$545$	−3794.86	−0.298264
$546$	0	0
$547$	−14925.2	−1.16665	−0.583325	−	0.812239i	$-0.698248\pi$
$547$	−14925.2	−1.16665	−0.583325	+	0.812239i	$0.698248\pi$
$548$	0	0
$549$	−43394.7	−3.37348
$550$	0	0
$551$	−2605.05	−0.201413
$552$	0	0
$553$	−5684.40	−0.437117
$554$	0	0
$555$	−2501.08	−0.191288
$556$	0	0
$557$	12187.8	0.927136	0.463568	−	0.886061i	$-0.346569\pi$
$557$	12187.8	0.927136	0.463568	+	0.886061i	$0.346569\pi$
$558$	0	0
$559$	−22179.6	−1.67817
$560$	0	0
$561$	217.207	0.0163467
$562$	0	0
$563$	−25198.0	−1.88627	−0.943133	−	0.332415i	$-0.892136\pi$
$563$	−25198.0	−1.88627	−0.943133	+	0.332415i	$0.892136\pi$
$564$	0	0
$565$	−3805.47	−0.283358
$566$	0	0
$567$	22068.7	1.63457
$568$	0	0
$569$	−14101.5	−1.03895	−0.519477	−	0.854485i	$-0.673873\pi$
$569$	−14101.5	−1.03895	−0.519477	+	0.854485i	$0.673873\pi$
$570$	0	0
$571$	21762.0	1.59494	0.797472	−	0.603356i	$-0.206170\pi$
$571$	21762.0	1.59494	0.797472	+	0.603356i	$0.206170\pi$
$572$	0	0
$573$	6511.31	0.474719
$574$	0	0
$575$	−13323.6	−0.966319
$576$	0	0
$577$	−17435.4	−1.25796	−0.628982	−	0.777420i	$-0.716528\pi$
$577$	−17435.4	−1.25796	−0.628982	+	0.777420i	$0.716528\pi$
$578$	0	0
$579$	9225.63	0.662183
$580$	0	0
$581$	−26379.3	−1.88365
$582$	0	0
$583$	−182.193	−0.0129428
$584$	0	0
$585$	−7724.53	−0.545932
$586$	0	0
$587$	−18080.4	−1.27131	−0.635655	−	0.771974i	$-0.719270\pi$
$587$	−18080.4	−1.27131	−0.635655	+	0.771974i	$0.719270\pi$
$588$	0	0
$589$	−23198.9	−1.62291
$590$	0	0
$591$	1649.16	0.114784
$592$	0	0
$593$	−24509.3	−1.69726	−0.848630	−	0.528987i	$-0.822572\pi$
$593$	−24509.3	−1.69726	−0.848630	+	0.528987i	$0.822572\pi$
$594$	0	0
$595$	−4096.37	−0.282243
$596$	0	0
$597$	−28494.0	−1.95340
$598$	0	0
$599$	−9457.05	−0.645083	−0.322541	−	0.946555i	$-0.604537\pi$
$599$	−9457.05	−0.645083	−0.322541	+	0.946555i	$0.604537\pi$
$600$	0	0
$601$	−11759.8	−0.798158	−0.399079	−	0.916917i	$-0.630670\pi$
$601$	−11759.8	−0.798158	−0.399079	+	0.916917i	$0.630670\pi$
$602$	0	0
$603$	15743.7	1.06324
$604$	0	0
$605$	−2870.33	−0.192885
$606$	0	0
$607$	17929.0	1.19887	0.599435	−	0.800424i	$-0.295392\pi$
$607$	17929.0	1.19887	0.599435	+	0.800424i	$0.295392\pi$
$608$	0	0
$609$	−7290.45	−0.485097
$610$	0	0
$611$	9638.43	0.638182
$612$	0	0
$613$	334.350	0.0220298	0.0110149	−	0.999939i	$-0.496494\pi$
$613$	334.350	0.0220298	0.0110149	+	0.999939i	$0.496494\pi$
$614$	0	0
$615$	5520.12	0.361939
$616$	0	0
$617$	23637.6	1.54233	0.771163	−	0.636638i	$-0.219675\pi$
$617$	23637.6	1.54233	0.771163	+	0.636638i	$0.219675\pi$
$618$	0	0
$619$	12855.8	0.834763	0.417381	−	0.908731i	$-0.362948\pi$
$619$	12855.8	0.834763	0.417381	+	0.908731i	$0.362948\pi$
$620$	0	0
$621$	27828.5	1.79826
$622$	0	0
$623$	38638.4	2.48477
$624$	0	0
$625$	13902.3	0.889750
$626$	0	0
$627$	−285.550	−0.0181878
$628$	0	0
$629$	8761.58	0.555401
$630$	0	0
$631$	−15364.3	−0.969321	−0.484660	−	0.874702i	$-0.661057\pi$
$631$	−15364.3	−0.969321	−0.484660	+	0.874702i	$0.661057\pi$
$632$	0	0
$633$	18344.5	1.15186
$634$	0	0
$635$	−2418.31	−0.151130
$636$	0	0
$637$	−28085.3	−1.74691
$638$	0	0
$639$	−29963.2	−1.85497
$640$	0	0
$641$	4238.75	0.261186	0.130593	−	0.991436i	$-0.458312\pi$
$641$	4238.75	0.261186	0.130593	+	0.991436i	$0.458312\pi$
$642$	0	0
$643$	−2694.68	−0.165269	−0.0826345	−	0.996580i	$-0.526333\pi$
$643$	−2694.68	−0.165269	−0.0826345	+	0.996580i	$0.526333\pi$
$644$	0	0
$645$	6618.53	0.404037
$646$	0	0
$647$	4728.45	0.287318	0.143659	−	0.989627i	$-0.454113\pi$
$647$	4728.45	0.287318	0.143659	+	0.989627i	$0.454113\pi$
$648$	0	0
$649$	35.9733	0.00217577
$650$	0	0
$651$	−64924.3	−3.90873
$652$	0	0
$653$	−2902.87	−0.173963	−0.0869817	−	0.996210i	$-0.527722\pi$
$653$	−2902.87	−0.173963	−0.0869817	+	0.996210i	$0.527722\pi$
$654$	0	0
$655$	2752.66	0.164207
$656$	0	0
$657$	−14254.3	−0.846440
$658$	0	0
$659$	3791.51	0.224122	0.112061	−	0.993701i	$-0.464255\pi$
$659$	3791.51	0.224122	0.112061	+	0.993701i	$0.464255\pi$
$660$	0	0
$661$	7566.63	0.445246	0.222623	−	0.974905i	$-0.428538\pi$
$661$	7566.63	0.445246	0.222623	+	0.974905i	$0.428538\pi$
$662$	0	0
$663$	40393.9	2.36617
$664$	0	0
$665$	5385.27	0.314033
$666$	0	0
$667$	−3210.55	−0.186376
$668$	0	0
$669$	−17565.4	−1.01513
$670$	0	0
$671$	278.362	0.0160150
$672$	0	0
$673$	18220.4	1.04360	0.521801	−	0.853067i	$-0.325260\pi$
$673$	18220.4	1.04360	0.521801	+	0.853067i	$0.325260\pi$
$674$	0	0
$675$	−30251.6	−1.72502
$676$	0	0
$677$	24827.8	1.40947	0.704735	−	0.709470i	$-0.251066\pi$
$677$	24827.8	1.40947	0.704735	+	0.709470i	$0.251066\pi$
$678$	0	0
$679$	34143.0	1.92973
$680$	0	0
$681$	31440.6	1.76917
$682$	0	0
$683$	−7554.87	−0.423249	−0.211625	−	0.977351i	$-0.567875\pi$
$683$	−7554.87	−0.423249	−0.211625	+	0.977351i	$0.567875\pi$
$684$	0	0
$685$	−3374.16	−0.188205
$686$	0	0
$687$	27270.3	1.51445
$688$	0	0
$689$	−33882.4	−1.87346
$690$	0	0
$691$	−11730.1	−0.645780	−0.322890	−	0.946437i	$-0.604654\pi$
$691$	−11730.1	−0.645780	−0.322890	+	0.946437i	$0.604654\pi$
$692$	0	0
$693$	−535.343	−0.0293449
$694$	0	0
$695$	4813.43	0.262711
$696$	0	0
$697$	−19337.6	−1.05088
$698$	0	0
$699$	49834.2	2.69657
$700$	0	0
$701$	−13791.3	−0.743065	−0.371533	−	0.928420i	$-0.621168\pi$
$701$	−13791.3	−0.743065	−0.371533	+	0.928420i	$0.621168\pi$
$702$	0	0
$703$	−11518.4	−0.617957
$704$	0	0
$705$	−2876.16	−0.153649
$706$	0	0
$707$	30305.5	1.61210
$708$	0	0
$709$	3236.05	0.171414	0.0857068	−	0.996320i	$-0.472685\pi$
$709$	3236.05	0.171414	0.0857068	+	0.996320i	$0.472685\pi$
$710$	0	0
$711$	−11205.2	−0.591038
$712$	0	0
$713$	−28591.2	−1.50175
$714$	0	0
$715$	49.5501	0.00259171
$716$	0	0
$717$	15763.0	0.821030
$718$	0	0
$719$	−5692.08	−0.295241	−0.147621	−	0.989044i	$-0.547161\pi$
$719$	−5692.08	−0.295241	−0.147621	+	0.989044i	$0.547161\pi$
$720$	0	0
$721$	−34105.4	−1.76165
$722$	0	0
$723$	−10750.1	−0.552975
$724$	0	0
$725$	3490.11	0.178785
$726$	0	0
$727$	−24211.1	−1.23513	−0.617565	−	0.786520i	$-0.711881\pi$
$727$	−24211.1	−1.23513	−0.617565	+	0.786520i	$0.711881\pi$
$728$	0	0
$729$	−17878.4	−0.908318
$730$	0	0
$731$	−23185.5	−1.17311
$732$	0	0
$733$	35957.5	1.81190	0.905948	−	0.423390i	$-0.139160\pi$
$733$	35957.5	1.81190	0.905948	+	0.423390i	$0.139160\pi$
$734$	0	0
$735$	8380.82	0.420587
$736$	0	0
$737$	−100.990	−0.00504752
$738$	0	0
$739$	15533.2	0.773206	0.386603	−	0.922246i	$-0.373648\pi$
$739$	15533.2	0.773206	0.386603	+	0.922246i	$0.373648\pi$
$740$	0	0
$741$	−53103.7	−2.63267
$742$	0	0
$743$	7606.78	0.375593	0.187797	−	0.982208i	$-0.439865\pi$
$743$	7606.78	0.375593	0.187797	+	0.982208i	$0.439865\pi$
$744$	0	0
$745$	−7494.50	−0.368560
$746$	0	0
$747$	−51999.5	−2.54694
$748$	0	0
$749$	−10154.9	−0.495395
$750$	0	0
$751$	−29108.6	−1.41437	−0.707183	−	0.707030i	$-0.750035\pi$
$751$	−29108.6	−1.41437	−0.707183	+	0.707030i	$0.750035\pi$
$752$	0	0
$753$	42321.1	2.04816
$754$	0	0
$755$	2289.58	0.110366
$756$	0	0
$757$	3291.98	0.158057	0.0790284	−	0.996872i	$-0.474818\pi$
$757$	3291.98	0.158057	0.0790284	+	0.996872i	$0.474818\pi$
$758$	0	0
$759$	−351.922	−0.0168300
$760$	0	0
$761$	−4341.98	−0.206829	−0.103414	−	0.994638i	$-0.532977\pi$
$761$	−4341.98	−0.206829	−0.103414	+	0.994638i	$0.532977\pi$
$762$	0	0
$763$	−48910.0	−2.32066
$764$	0	0
$765$	−8074.84	−0.381629
$766$	0	0
$767$	6689.95	0.314941
$768$	0	0
$769$	13820.2	0.648075	0.324038	−	0.946044i	$-0.394960\pi$
$769$	13820.2	0.648075	0.324038	+	0.946044i	$0.394960\pi$
$770$	0	0
$771$	20108.6	0.939292
$772$	0	0
$773$	19762.4	0.919539	0.459769	−	0.888038i	$-0.347932\pi$
$773$	19762.4	0.919539	0.459769	+	0.888038i	$0.347932\pi$
$774$	0	0
$775$	31080.8	1.44059
$776$	0	0
$777$	−32235.2	−1.48833
$778$	0	0
$779$	25422.1	1.16924
$780$	0	0
$781$	192.204	0.00880613
$782$	0	0
$783$	−7289.64	−0.332708
$784$	0	0
$785$	−3589.64	−0.163210
$786$	0	0
$787$	12090.5	0.547622	0.273811	−	0.961784i	$-0.411716\pi$
$787$	12090.5	0.547622	0.273811	+	0.961784i	$0.411716\pi$
$788$	0	0
$789$	60784.2	2.74268
$790$	0	0
$791$	−49046.8	−2.20468
$792$	0	0
$793$	51766.9	2.31815
$794$	0	0
$795$	10110.7	0.451056
$796$	0	0
$797$	−26490.3	−1.17733	−0.588667	−	0.808376i	$-0.700347\pi$
$797$	−26490.3	−1.17733	−0.588667	+	0.808376i	$0.700347\pi$
$798$	0	0
$799$	10075.5	0.446116
$800$	0	0
$801$	76164.7	3.35973
$802$	0	0
$803$	91.4360	0.00401832
$804$	0	0
$805$	6637.00	0.290588
$806$	0	0
$807$	−25710.7	−1.12151
$808$	0	0
$809$	−21825.5	−0.948508	−0.474254	−	0.880388i	$-0.657282\pi$
$809$	−21825.5	−0.948508	−0.474254	+	0.880388i	$0.657282\pi$
$810$	0	0
$811$	28758.7	1.24520	0.622599	−	0.782541i	$-0.286077\pi$
$811$	28758.7	1.24520	0.622599	+	0.782541i	$0.286077\pi$
$812$	0	0
$813$	−34304.0	−1.47982
$814$	0	0
$815$	−515.809	−0.0221693
$816$	0	0
$817$	30480.7	1.30524
$818$	0	0
$819$	−99557.6	−4.24765
$820$	0	0
$821$	3491.27	0.148412	0.0742060	−	0.997243i	$-0.476358\pi$
$821$	3491.27	0.148412	0.0742060	+	0.997243i	$0.476358\pi$
$822$	0	0
$823$	−1472.33	−0.0623597	−0.0311799	−	0.999514i	$-0.509926\pi$
$823$	−1472.33	−0.0623597	−0.0311799	+	0.999514i	$0.509926\pi$
$824$	0	0
$825$	382.565	0.0161445
$826$	0	0
$827$	−17843.8	−0.750289	−0.375145	−	0.926966i	$-0.622407\pi$
$827$	−17843.8	−0.750289	−0.375145	+	0.926966i	$0.622407\pi$
$828$	0	0
$829$	19574.7	0.820093	0.410047	−	0.912065i	$-0.365512\pi$
$829$	19574.7	0.820093	0.410047	+	0.912065i	$0.365512\pi$
$830$	0	0
$831$	60578.4	2.52881
$832$	0	0
$833$	−29359.0	−1.22116
$834$	0	0
$835$	2869.36	0.118920
$836$	0	0
$837$	−64917.0	−2.68084
$838$	0	0
$839$	−26014.0	−1.07045	−0.535223	−	0.844711i	$-0.679772\pi$
$839$	−26014.0	−1.07045	−0.535223	+	0.844711i	$0.679772\pi$
$840$	0	0
$841$	841.000	0.0344828
$842$	0	0
$843$	77171.9	3.15296
$844$	0	0
$845$	4476.52	0.182245
$846$	0	0
$847$	−36994.2	−1.50075
$848$	0	0
$849$	−82307.8	−3.32721
$850$	0	0
$851$	−14195.6	−0.571822
$852$	0	0
$853$	17613.3	0.706996	0.353498	−	0.935435i	$-0.384992\pi$
$853$	17613.3	0.706996	0.353498	+	0.935435i	$0.384992\pi$
$854$	0	0
$855$	10615.6	0.424613
$856$	0	0
$857$	−40047.5	−1.59626	−0.798130	−	0.602485i	$-0.794177\pi$
$857$	−40047.5	−1.59626	−0.798130	+	0.602485i	$0.794177\pi$
$858$	0	0
$859$	10686.0	0.424448	0.212224	−	0.977221i	$-0.431929\pi$
$859$	10686.0	0.424448	0.212224	+	0.977221i	$0.431929\pi$
$860$	0	0
$861$	71146.0	2.81609
$862$	0	0
$863$	24243.4	0.956263	0.478131	−	0.878288i	$-0.341314\pi$
$863$	24243.4	0.956263	0.478131	+	0.878288i	$0.341314\pi$
$864$	0	0
$865$	6486.02	0.254950
$866$	0	0
$867$	−2207.36	−0.0864656
$868$	0	0
$869$	71.8775	0.00280584
$870$	0	0
$871$	−18781.1	−0.730626
$872$	0	0
$873$	67303.3	2.60924
$874$	0	0
$875$	−14708.7	−0.568279
$876$	0	0
$877$	32542.0	1.25298	0.626491	−	0.779429i	$-0.284491\pi$
$877$	32542.0	1.25298	0.626491	+	0.779429i	$0.284491\pi$
$878$	0	0
$879$	58029.2	2.22671
$880$	0	0
$881$	28538.8	1.09137	0.545686	−	0.837990i	$-0.316269\pi$
$881$	28538.8	1.09137	0.545686	+	0.837990i	$0.316269\pi$
$882$	0	0
$883$	−3707.98	−0.141317	−0.0706587	−	0.997501i	$-0.522510\pi$
$883$	−3707.98	−0.141317	−0.0706587	+	0.997501i	$0.522510\pi$
$884$	0	0
$885$	−1996.32	−0.0758254
$886$	0	0
$887$	4461.95	0.168904	0.0844519	−	0.996428i	$-0.473086\pi$
$887$	4461.95	0.168904	0.0844519	+	0.996428i	$0.473086\pi$
$888$	0	0
$889$	−31168.4	−1.17588
$890$	0	0
$891$	−279.052	−0.0104922
$892$	0	0
$893$	−13245.8	−0.496363
$894$	0	0
$895$	1557.78	0.0581798
$896$	0	0
$897$	−65446.8	−2.43613
$898$	0	0
$899$	7489.43	0.277849
$900$	0	0
$901$	−35418.9	−1.30963
$902$	0	0
$903$	85302.9	3.14363
$904$	0	0
$905$	5960.99	0.218950
$906$	0	0
$907$	30719.9	1.12463	0.562313	−	0.826924i	$-0.309912\pi$
$907$	30719.9	1.12463	0.562313	+	0.826924i	$0.309912\pi$
$908$	0	0
$909$	59738.9	2.17977
$910$	0	0
$911$	22983.9	0.835886	0.417943	−	0.908473i	$-0.362751\pi$
$911$	22983.9	0.835886	0.417943	+	0.908473i	$0.362751\pi$
$912$	0	0
$913$	333.558	0.0120911
$914$	0	0
$915$	−15447.5	−0.558120
$916$	0	0
$917$	35477.7	1.27762
$918$	0	0
$919$	−52064.7	−1.86883	−0.934416	−	0.356183i	$-0.884078\pi$
$919$	−52064.7	−1.86883	−0.934416	+	0.356183i	$0.884078\pi$
$920$	0	0
$921$	88957.8	3.18269
$922$	0	0
$923$	35744.1	1.27468
$924$	0	0
$925$	15431.7	0.548532
$926$	0	0
$927$	−67229.3	−2.38199
$928$	0	0
$929$	−48448.1	−1.71101	−0.855506	−	0.517793i	$-0.826754\pi$
$929$	−48448.1	−1.71101	−0.855506	+	0.517793i	$0.826754\pi$
$930$	0	0
$931$	38596.7	1.35871
$932$	0	0
$933$	61562.0	2.16018
$934$	0	0
$935$	51.7973	0.00181171
$936$	0	0
$937$	20972.3	0.731199	0.365600	−	0.930772i	$-0.380864\pi$
$937$	20972.3	0.731199	0.365600	+	0.930772i	$0.380864\pi$
$938$	0	0
$939$	16822.2	0.584633
$940$	0	0
$941$	−19883.9	−0.688840	−0.344420	−	0.938816i	$-0.611924\pi$
$941$	−19883.9	−0.688840	−0.344420	+	0.938816i	$0.611924\pi$
$942$	0	0
$943$	31331.1	1.08195
$944$	0	0
$945$	15069.5	0.518741
$946$	0	0
$947$	−45811.5	−1.57199	−0.785995	−	0.618233i	$-0.787849\pi$
$947$	−45811.5	−1.57199	−0.785995	+	0.618233i	$0.787849\pi$
$948$	0	0
$949$	17004.3	0.581649
$950$	0	0
$951$	21040.9	0.717454
$952$	0	0
$953$	−24303.7	−0.826101	−0.413050	−	0.910708i	$-0.635537\pi$
$953$	−24303.7	−0.826101	−0.413050	+	0.910708i	$0.635537\pi$
$954$	0	0
$955$	1552.75	0.0526135
$956$	0	0
$957$	92.1854	0.00311383
$958$	0	0
$959$	−43487.9	−1.46434
$960$	0	0
$961$	36905.3	1.23881
$962$	0	0
$963$	−20017.5	−0.669838
$964$	0	0
$965$	2200.04	0.0733903
$966$	0	0
$967$	46462.1	1.54511	0.772555	−	0.634948i	$-0.218979\pi$
$967$	46462.1	1.54511	0.772555	+	0.634948i	$0.218979\pi$
$968$	0	0
$969$	−55511.9	−1.84035
$970$	0	0
$971$	−5526.77	−0.182660	−0.0913298	−	0.995821i	$-0.529112\pi$
$971$	−5526.77	−0.182660	−0.0913298	+	0.995821i	$0.529112\pi$
$972$	0	0
$973$	62037.9	2.04403
$974$	0	0
$975$	71145.6	2.33690
$976$	0	0
$977$	12957.3	0.424300	0.212150	−	0.977237i	$-0.431953\pi$
$977$	12957.3	0.424300	0.212150	+	0.977237i	$0.431953\pi$
$978$	0	0
$979$	−488.570	−0.0159497
$980$	0	0
$981$	−96412.4	−3.13783
$982$	0	0
$983$	57988.1	1.88152	0.940759	−	0.339076i	$-0.110114\pi$
$983$	57988.1	1.88152	0.940759	+	0.339076i	$0.110114\pi$
$984$	0	0
$985$	393.275	0.0127216
$986$	0	0
$987$	−37069.4	−1.19547
$988$	0	0
$989$	37565.5	1.20780
$990$	0	0
$991$	−33344.7	−1.06885	−0.534425	−	0.845216i	$-0.679472\pi$
$991$	−33344.7	−1.06885	−0.534425	+	0.845216i	$0.679472\pi$
$992$	0	0
$993$	−15094.2	−0.482378
$994$	0	0
$995$	−6794.96	−0.216497
$996$	0	0
$997$	−13474.8	−0.428034	−0.214017	−	0.976830i	$-0.568655\pi$
$997$	−13474.8	−0.428034	−0.214017	+	0.976830i	$0.568655\pi$
$998$	0	0
$999$	−32231.6	−1.02078

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	232.4.a.b.1.3	✓	3
3.2	odd	2		2088.4.a.b.1.2		3
4.3	odd	2		464.4.a.g.1.1		3
8.3	odd	2		1856.4.a.u.1.3		3
8.5	even	2		1856.4.a.p.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
232.4.a.b.1.3	✓	3	1.1	even	1	trivial
464.4.a.g.1.1		3	4.3	odd	2
1856.4.a.p.1.1		3	8.5	even	2
1856.4.a.u.1.3		3	8.3	odd	2
2088.4.a.b.1.2		3	3.2	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:	\( N \)	\(=\)	\( 232 = 2^{3} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	232.a (trivial)

\(f(q)\)	\(=\)	\(q+9.04399 q^{3} +2.15672 q^{5} +27.7969 q^{7} +54.7938 q^{9} +O(q^{10})\)	\(q+9.04399 q^{3} +2.15672 q^{5} +27.7969 q^{7} +54.7938 q^{9} -0.351483 q^{11} -65.3652 q^{13} +19.5054 q^{15} -68.3296 q^{17} +89.8292 q^{19} +251.395 q^{21} +110.709 q^{23} -120.349 q^{25} +251.367 q^{27} -29.0000 q^{29} -258.256 q^{31} -3.17881 q^{33} +59.9501 q^{35} -128.225 q^{37} -591.163 q^{39} +283.005 q^{41} +339.318 q^{43} +118.175 q^{45} -147.455 q^{47} +429.668 q^{49} -617.972 q^{51} +518.354 q^{53} -0.758050 q^{55} +812.414 q^{57} -102.347 q^{59} -791.964 q^{61} +1523.10 q^{63} -140.975 q^{65} +287.326 q^{67} +1001.25 q^{69} -546.837 q^{71} -260.144 q^{73} -1088.43 q^{75} -9.77013 q^{77} -204.498 q^{79} +793.927 q^{81} -949.003 q^{83} -147.368 q^{85} -262.276 q^{87} +1390.02 q^{89} -1816.95 q^{91} -2335.67 q^{93} +193.736 q^{95} +1228.30 q^{97} -19.2591 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 6 q^{3} + 4 q^{5} + 16 q^{7} + 45 q^{9}+O(q^{10}) \) 3 * q + 6 * q^3 + 4 * q^5 + 16 * q^7 + 45 * q^9	\( 3 q + 6 q^{3} + 4 q^{5} + 16 q^{7} + 45 q^{9} - 2 q^{11} + 28 q^{13} + 136 q^{15} - 66 q^{17} - 66 q^{19} + 472 q^{21} + 176 q^{23} - 9 q^{25} + 228 q^{27} - 87 q^{29} - 190 q^{31} + 154 q^{33} + 660 q^{35} + 442 q^{37} - 656 q^{39} + 1162 q^{41} + 30 q^{43} - 254 q^{45} - 738 q^{47} + 851 q^{49} - 576 q^{51} + 312 q^{53} + 464 q^{55} + 684 q^{57} + 44 q^{59} + 54 q^{61} + 964 q^{63} + 178 q^{65} - 116 q^{67} + 812 q^{69} - 1200 q^{71} - 1118 q^{73} - 1038 q^{75} + 792 q^{77} - 2262 q^{79} + 15 q^{81} - 1804 q^{83} - 8 q^{85} - 174 q^{87} + 1578 q^{89} - 1972 q^{91} - 706 q^{93} - 1052 q^{95} + 1450 q^{97} - 482 q^{99}+O(q^{100}) \) 3 * q + 6 * q^3 + 4 * q^5 + 16 * q^7 + 45 * q^9 - 2 * q^11 + 28 * q^13 + 136 * q^15 - 66 * q^17 - 66 * q^19 + 472 * q^21 + 176 * q^23 - 9 * q^25 + 228 * q^27 - 87 * q^29 - 190 * q^31 + 154 * q^33 + 660 * q^35 + 442 * q^37 - 656 * q^39 + 1162 * q^41 + 30 * q^43 - 254 * q^45 - 738 * q^47 + 851 * q^49 - 576 * q^51 + 312 * q^53 + 464 * q^55 + 684 * q^57 + 44 * q^59 + 54 * q^61 + 964 * q^63 + 178 * q^65 - 116 * q^67 + 812 * q^69 - 1200 * q^71 - 1118 * q^73 - 1038 * q^75 + 792 * q^77 - 2262 * q^79 + 15 * q^81 - 1804 * q^83 - 8 * q^85 - 174 * q^87 + 1578 * q^89 - 1972 * q^91 - 706 * q^93 - 1052 * q^95 + 1450 * q^97 - 482 * q^99

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