LMFDB - Embedded newform 464.4.a.f.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [464,4,Mod(1,464)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(464, 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("464.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 464 = 2^{4} \cdot 29 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	464.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:	$27.3768862427$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 29)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.41421$ of defining polynomial
Character	$\chi$	$=$	464.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.757359 q^{3} -10.6569 q^{5} +22.1421 q^{7} -26.4264 q^{9} +O(q^{10})$	$q+0.757359 q^{3} -10.6569 q^{5} +22.1421 q^{7} -26.4264 q^{9} -39.3259 q^{11} +23.7696 q^{13} -8.07107 q^{15} +4.54416 q^{17} +155.255 q^{19} +16.7696 q^{21} +41.8823 q^{23} -11.4315 q^{25} -40.4630 q^{27} +29.0000 q^{29} +57.9045 q^{31} -29.7838 q^{33} -235.966 q^{35} +235.196 q^{37} +18.0021 q^{39} -175.161 q^{41} +402.831 q^{43} +281.622 q^{45} -227.742 q^{47} +147.274 q^{49} +3.44156 q^{51} +673.534 q^{53} +419.090 q^{55} +117.584 q^{57} +800.725 q^{59} -222.270 q^{61} -585.137 q^{63} -253.309 q^{65} +524.479 q^{67} +31.7199 q^{69} +281.917 q^{71} +1229.10 q^{73} -8.65772 q^{75} -870.759 q^{77} -611.247 q^{79} +682.868 q^{81} -515.490 q^{83} -48.4264 q^{85} +21.9634 q^{87} -358.219 q^{89} +526.309 q^{91} +43.8545 q^{93} -1654.53 q^{95} +829.415 q^{97} +1039.24 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 10 q^{3} - 10 q^{5} + 16 q^{7} + 32 q^{9}+O(q^{10}) $ 2 * q + 10 * q^3 - 10 * q^5 + 16 * q^7 + 32 * q^9	$ 2 q + 10 q^{3} - 10 q^{5} + 16 q^{7} + 32 q^{9} + 26 q^{11} - 26 q^{13} - 2 q^{15} + 60 q^{17} + 220 q^{19} - 40 q^{21} - 52 q^{23} - 136 q^{25} + 250 q^{27} + 58 q^{29} + 294 q^{31} + 574 q^{33} - 240 q^{35} + 312 q^{37} - 442 q^{39} + 40 q^{41} + 322 q^{43} + 320 q^{45} + 130 q^{47} - 158 q^{49} + 516 q^{51} + 1002 q^{53} + 462 q^{55} + 716 q^{57} + 900 q^{59} - 948 q^{61} - 944 q^{63} - 286 q^{65} - 320 q^{67} - 836 q^{69} + 660 q^{71} + 648 q^{73} - 1160 q^{75} - 1272 q^{77} - 258 q^{79} + 1790 q^{81} - 1212 q^{83} - 12 q^{85} + 290 q^{87} + 760 q^{89} + 832 q^{91} + 2226 q^{93} - 1612 q^{95} + 24 q^{97} + 4856 q^{99}+O(q^{100}) $ 2 * q + 10 * q^3 - 10 * q^5 + 16 * q^7 + 32 * q^9 + 26 * q^11 - 26 * q^13 - 2 * q^15 + 60 * q^17 + 220 * q^19 - 40 * q^21 - 52 * q^23 - 136 * q^25 + 250 * q^27 + 58 * q^29 + 294 * q^31 + 574 * q^33 - 240 * q^35 + 312 * q^37 - 442 * q^39 + 40 * q^41 + 322 * q^43 + 320 * q^45 + 130 * q^47 - 158 * q^49 + 516 * q^51 + 1002 * q^53 + 462 * q^55 + 716 * q^57 + 900 * q^59 - 948 * q^61 - 944 * q^63 - 286 * q^65 - 320 * q^67 - 836 * q^69 + 660 * q^71 + 648 * q^73 - 1160 * q^75 - 1272 * q^77 - 258 * q^79 + 1790 * q^81 - 1212 * q^83 - 12 * q^85 + 290 * q^87 + 760 * q^89 + 832 * q^91 + 2226 * q^93 - 1612 * q^95 + 24 * q^97 + 4856 * 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
$3$	0.757359	0.145754	0.0728769	−	0.997341i	$-0.476782\pi$
$3$	0.757359	0.145754	0.0728769	+	0.997341i	$0.476782\pi$
$4$	0	0
$5$	−10.6569	−0.953178	−0.476589	−	0.879126i	$-0.658127\pi$
$5$	−10.6569	−0.953178	−0.476589	+	0.879126i	$0.658127\pi$
$6$	0	0
$7$	22.1421	1.19556	0.597781	−	0.801659i	$-0.296049\pi$
$7$	22.1421	1.19556	0.597781	+	0.801659i	$0.296049\pi$
$8$	0	0
$9$	−26.4264	−0.978756
$10$	0	0
$11$	−39.3259	−1.07793	−0.538964	−	0.842329i	$-0.681184\pi$
$11$	−39.3259	−1.07793	−0.538964	+	0.842329i	$0.681184\pi$
$12$	0	0
$13$	23.7696	0.507114	0.253557	−	0.967320i	$-0.418399\pi$
$13$	23.7696	0.507114	0.253557	+	0.967320i	$0.418399\pi$
$14$	0	0
$15$	−8.07107	−0.138929
$16$	0	0
$17$	4.54416	0.0648306	0.0324153	−	0.999474i	$-0.489680\pi$
$17$	4.54416	0.0648306	0.0324153	+	0.999474i	$0.489680\pi$
$18$	0	0
$19$	155.255	1.87463	0.937313	−	0.348488i	$-0.113305\pi$
$19$	155.255	1.87463	0.937313	+	0.348488i	$0.113305\pi$
$20$	0	0
$21$	16.7696	0.174258
$22$	0	0
$23$	41.8823	0.379698	0.189849	−	0.981813i	$-0.439200\pi$
$23$	41.8823	0.379698	0.189849	+	0.981813i	$0.439200\pi$
$24$	0	0
$25$	−11.4315	−0.0914517
$26$	0	0
$27$	−40.4630	−0.288411
$28$	0	0
$29$	29.0000	0.185695
$29$	29.0000	0.185695
$30$	0	0
$31$	57.9045	0.335483	0.167741	−	0.985831i	$-0.446353\pi$
$31$	57.9045	0.335483	0.167741	+	0.985831i	$0.446353\pi$
$32$	0	0
$33$	−29.7838	−0.157112
$34$	0	0
$35$	−235.966	−1.13958
$36$	0	0
$37$	235.196	1.04503	0.522513	−	0.852631i	$-0.324995\pi$
$37$	235.196	1.04503	0.522513	+	0.852631i	$0.324995\pi$
$38$	0	0
$39$	18.0021	0.0739139
$40$	0	0
$41$	−175.161	−0.667210	−0.333605	−	0.942713i	$-0.608265\pi$
$41$	−175.161	−0.667210	−0.333605	+	0.942713i	$0.608265\pi$
$42$	0	0
$43$	402.831	1.42863	0.714315	−	0.699824i	$-0.246738\pi$
$43$	402.831	1.42863	0.714315	+	0.699824i	$0.246738\pi$
$44$	0	0
$45$	281.622	0.932929
$46$	0	0
$47$	−227.742	−0.706800	−0.353400	−	0.935472i	$-0.614975\pi$
$47$	−227.742	−0.706800	−0.353400	+	0.935472i	$0.614975\pi$
$48$	0	0
$49$	147.274	0.429371
$50$	0	0
$51$	3.44156	0.00944931
$52$	0	0
$53$	673.534	1.74560	0.872802	−	0.488074i	$-0.162301\pi$
$53$	673.534	1.74560	0.872802	+	0.488074i	$0.162301\pi$
$54$	0	0
$55$	419.090	1.02746
$56$	0	0
$57$	117.584	0.273234
$58$	0	0
$59$	800.725	1.76687	0.883437	−	0.468551i	$-0.155224\pi$
$59$	800.725	1.76687	0.883437	+	0.468551i	$0.155224\pi$
$60$	0	0
$61$	−222.270	−0.466537	−0.233268	−	0.972412i	$-0.574942\pi$
$61$	−222.270	−0.466537	−0.233268	+	0.972412i	$0.574942\pi$
$62$	0	0
$63$	−585.137	−1.17016
$64$	0	0
$65$	−253.309	−0.483370
$66$	0	0
$67$	524.479	0.956349	0.478174	−	0.878265i	$-0.341299\pi$
$67$	524.479	0.956349	0.478174	+	0.878265i	$0.341299\pi$
$68$	0	0
$69$	31.7199	0.0553424
$70$	0	0
$71$	281.917	0.471230	0.235615	−	0.971846i	$-0.424289\pi$
$71$	281.917	0.471230	0.235615	+	0.971846i	$0.424289\pi$
$72$	0	0
$73$	1229.10	1.97061	0.985307	−	0.170790i	$-0.0546320\pi$
$73$	1229.10	1.97061	0.985307	+	0.170790i	$0.0546320\pi$
$74$	0	0
$75$	−8.65772	−0.0133294
$76$	0	0
$77$	−870.759	−1.28873
$78$	0	0
$79$	−611.247	−0.870514	−0.435257	−	0.900306i	$-0.643343\pi$
$79$	−611.247	−0.870514	−0.435257	+	0.900306i	$0.643343\pi$
$80$	0	0
$81$	682.868	0.936719
$82$	0	0
$83$	−515.490	−0.681716	−0.340858	−	0.940115i	$-0.610718\pi$
$83$	−515.490	−0.681716	−0.340858	+	0.940115i	$0.610718\pi$
$84$	0	0
$85$	−48.4264	−0.0617951
$86$	0	0
$87$	21.9634	0.0270658
$88$	0	0
$89$	−358.219	−0.426643	−0.213321	−	0.976982i	$-0.568428\pi$
$89$	−358.219	−0.426643	−0.213321	+	0.976982i	$0.568428\pi$
$90$	0	0
$91$	526.309	0.606287
$92$	0	0
$93$	43.8545	0.0488979
$94$	0	0
$95$	−1654.53	−1.78685
$96$	0	0
$97$	829.415	0.868189	0.434095	−	0.900867i	$-0.357068\pi$
$97$	829.415	0.868189	0.434095	+	0.900867i	$0.357068\pi$
$98$	0	0
$99$	1039.24	1.05503
$100$	0	0
$101$	−978.010	−0.963521	−0.481761	−	0.876303i	$-0.660003\pi$
$101$	−978.010	−0.963521	−0.481761	+	0.876303i	$0.660003\pi$
$102$	0	0
$103$	1217.33	1.16453	0.582266	−	0.812998i	$-0.302166\pi$
$103$	1217.33	1.16453	0.582266	+	0.812998i	$0.302166\pi$
$104$	0	0
$105$	−178.711	−0.166099
$106$	0	0
$107$	707.044	0.638808	0.319404	−	0.947619i	$-0.396517\pi$
$107$	707.044	0.638808	0.319404	+	0.947619i	$0.396517\pi$
$108$	0	0
$109$	−1496.14	−1.31471	−0.657357	−	0.753580i	$-0.728326\pi$
$109$	−1496.14	−1.31471	−0.657357	+	0.753580i	$0.728326\pi$
$110$	0	0
$111$	178.128	0.152317
$112$	0	0
$113$	−1067.23	−0.888469	−0.444234	−	0.895911i	$-0.646524\pi$
$113$	−1067.23	−0.888469	−0.444234	+	0.895911i	$0.646524\pi$
$114$	0	0
$115$	−446.333	−0.361920
$116$	0	0
$117$	−628.144	−0.496341
$118$	0	0
$119$	100.617	0.0775090
$120$	0	0
$121$	215.527	0.161928
$122$	0	0
$123$	−132.660	−0.0972485
$124$	0	0
$125$	1453.93	1.04035
$126$	0	0
$127$	1179.58	0.824177	0.412088	−	0.911144i	$-0.364799\pi$
$127$	1179.58	0.824177	0.412088	+	0.911144i	$0.364799\pi$
$128$	0	0
$129$	305.087	0.208228
$130$	0	0
$131$	2357.47	1.57231	0.786156	−	0.618028i	$-0.212068\pi$
$131$	2357.47	1.57231	0.786156	+	0.618028i	$0.212068\pi$
$132$	0	0
$133$	3437.67	2.24123
$134$	0	0
$135$	431.208	0.274907
$136$	0	0
$137$	722.489	0.450558	0.225279	−	0.974294i	$-0.427671\pi$
$137$	722.489	0.450558	0.225279	+	0.974294i	$0.427671\pi$
$138$	0	0
$139$	−1398.24	−0.853219	−0.426610	−	0.904436i	$-0.640292\pi$
$139$	−1398.24	−0.853219	−0.426610	+	0.904436i	$0.640292\pi$
$140$	0	0
$141$	−172.483	−0.103019
$142$	0	0
$143$	−934.759	−0.546633
$144$	0	0
$145$	−309.049	−0.177001
$146$	0	0
$147$	111.539	0.0625824
$148$	0	0
$149$	2830.63	1.55634	0.778168	−	0.628056i	$-0.216149\pi$
$149$	2830.63	1.55634	0.778168	+	0.628056i	$0.216149\pi$
$150$	0	0
$151$	−1705.58	−0.919194	−0.459597	−	0.888128i	$-0.652006\pi$
$151$	−1705.58	−0.919194	−0.459597	+	0.888128i	$0.652006\pi$
$152$	0	0
$153$	−120.086	−0.0634533
$154$	0	0
$155$	−617.080	−0.319775
$156$	0	0
$157$	−2670.84	−1.35768	−0.678841	−	0.734286i	$-0.737517\pi$
$157$	−2670.84	−1.35768	−0.678841	+	0.734286i	$0.737517\pi$
$158$	0	0
$159$	510.107	0.254429
$160$	0	0
$161$	927.362	0.453953
$162$	0	0
$163$	2151.92	1.03406	0.517028	−	0.855968i	$-0.327038\pi$
$163$	2151.92	1.03406	0.517028	+	0.855968i	$0.327038\pi$
$164$	0	0
$165$	317.402	0.149756
$166$	0	0
$167$	−999.387	−0.463083	−0.231542	−	0.972825i	$-0.574377\pi$
$167$	−999.387	−0.463083	−0.231542	+	0.972825i	$0.574377\pi$
$168$	0	0
$169$	−1632.01	−0.742835
$170$	0	0
$171$	−4102.83	−1.83480
$172$	0	0
$173$	−2534.30	−1.11375	−0.556877	−	0.830595i	$-0.688001\pi$
$173$	−2534.30	−1.11375	−0.556877	+	0.830595i	$0.688001\pi$
$174$	0	0
$175$	−253.117	−0.109336
$176$	0	0
$177$	606.437	0.257529
$178$	0	0
$179$	−3550.27	−1.48245	−0.741227	−	0.671254i	$-0.765756\pi$
$179$	−3550.27	−1.48245	−0.741227	+	0.671254i	$0.765756\pi$
$180$	0	0
$181$	−3034.68	−1.24622	−0.623110	−	0.782135i	$-0.714131\pi$
$181$	−3034.68	−1.24622	−0.623110	+	0.782135i	$0.714131\pi$
$182$	0	0
$183$	−168.338	−0.0679996
$184$	0	0
$185$	−2506.45	−0.996096
$186$	0	0
$187$	−178.703	−0.0698827
$188$	0	0
$189$	−895.937	−0.344814
$190$	0	0
$191$	−2224.00	−0.842529	−0.421265	−	0.906938i	$-0.638414\pi$
$191$	−2224.00	−0.842529	−0.421265	+	0.906938i	$0.638414\pi$
$192$	0	0
$193$	−632.830	−0.236021	−0.118011	−	0.993012i	$-0.537652\pi$
$193$	−632.830	−0.236021	−0.118011	+	0.993012i	$0.537652\pi$
$194$	0	0
$195$	−191.846	−0.0704531
$196$	0	0
$197$	1369.80	0.495404	0.247702	−	0.968836i	$-0.420325\pi$
$197$	1369.80	0.495404	0.247702	+	0.968836i	$0.420325\pi$
$198$	0	0
$199$	−1416.38	−0.504544	−0.252272	−	0.967656i	$-0.581178\pi$
$199$	−1416.38	−0.504544	−0.252272	+	0.967656i	$0.581178\pi$
$200$	0	0
$201$	397.219	0.139391
$202$	0	0
$203$	642.122	0.222010
$204$	0	0
$205$	1866.67	0.635970
$206$	0	0
$207$	−1106.80	−0.371632
$208$	0	0
$209$	−6105.54	−2.02071
$210$	0	0
$211$	896.432	0.292478	0.146239	−	0.989249i	$-0.453283\pi$
$211$	896.432	0.292478	0.146239	+	0.989249i	$0.453283\pi$
$212$	0	0
$213$	213.512	0.0686837
$214$	0	0
$215$	−4292.91	−1.36174
$216$	0	0
$217$	1282.13	0.401091
$218$	0	0
$219$	930.868	0.287225
$220$	0	0
$221$	108.013	0.0328765
$222$	0	0
$223$	2268.94	0.681344	0.340672	−	0.940182i	$-0.389345\pi$
$223$	2268.94	0.681344	0.340672	+	0.940182i	$0.389345\pi$
$224$	0	0
$225$	302.092	0.0895088
$226$	0	0
$227$	2078.09	0.607610	0.303805	−	0.952734i	$-0.401743\pi$
$227$	2078.09	0.607610	0.303805	+	0.952734i	$0.401743\pi$
$228$	0	0
$229$	−3715.05	−1.07204	−0.536020	−	0.844205i	$-0.680073\pi$
$229$	−3715.05	−1.07204	−0.536020	+	0.844205i	$0.680073\pi$
$230$	0	0
$231$	−659.478	−0.187837
$232$	0	0
$233$	2521.35	0.708923	0.354461	−	0.935071i	$-0.384664\pi$
$233$	2521.35	0.708923	0.354461	+	0.935071i	$0.384664\pi$
$234$	0	0
$235$	2427.02	0.673707
$236$	0	0
$237$	−462.933	−0.126881
$238$	0	0
$239$	−3940.04	−1.06636	−0.533179	−	0.846002i	$-0.679003\pi$
$239$	−3940.04	−1.06636	−0.533179	+	0.846002i	$0.679003\pi$
$240$	0	0
$241$	−1973.06	−0.527369	−0.263684	−	0.964609i	$-0.584938\pi$
$241$	−1973.06	−0.527369	−0.263684	+	0.964609i	$0.584938\pi$
$242$	0	0
$243$	1609.68	0.424942
$244$	0	0
$245$	−1569.48	−0.409267
$246$	0	0
$247$	3690.34	0.950650
$248$	0	0
$249$	−390.411	−0.0993627
$250$	0	0
$251$	1236.65	0.310982	0.155491	−	0.987837i	$-0.450304\pi$
$251$	1236.65	0.310982	0.155491	+	0.987837i	$0.450304\pi$
$252$	0	0
$253$	−1647.06	−0.409287
$254$	0	0
$255$	−36.6762	−0.00900687
$256$	0	0
$257$	2918.10	0.708272	0.354136	−	0.935194i	$-0.384775\pi$
$257$	2918.10	0.708272	0.354136	+	0.935194i	$0.384775\pi$
$258$	0	0
$259$	5207.74	1.24939
$260$	0	0
$261$	−766.366	−0.181750
$262$	0	0
$263$	310.789	0.0728673	0.0364336	−	0.999336i	$-0.488400\pi$
$263$	310.789	0.0728673	0.0364336	+	0.999336i	$0.488400\pi$
$264$	0	0
$265$	−7177.75	−1.66387
$266$	0	0
$267$	−271.301	−0.0621848
$268$	0	0
$269$	−1839.98	−0.417047	−0.208523	−	0.978017i	$-0.566866\pi$
$269$	−1839.98	−0.417047	−0.208523	+	0.978017i	$0.566866\pi$
$270$	0	0
$271$	−5187.35	−1.16277	−0.581383	−	0.813630i	$-0.697488\pi$
$271$	−5187.35	−1.16277	−0.581383	+	0.813630i	$0.697488\pi$
$272$	0	0
$273$	398.605	0.0883687
$274$	0	0
$275$	449.552	0.0985783
$276$	0	0
$277$	8666.41	1.87983	0.939917	−	0.341403i	$-0.110902\pi$
$277$	8666.41	1.87983	0.939917	+	0.341403i	$0.110902\pi$
$278$	0	0
$279$	−1530.21	−0.328356
$280$	0	0
$281$	7856.96	1.66800	0.833998	−	0.551767i	$-0.186046\pi$
$281$	7856.96	1.66800	0.833998	+	0.551767i	$0.186046\pi$
$282$	0	0
$283$	3054.71	0.641638	0.320819	−	0.947141i	$-0.396042\pi$
$283$	3054.71	0.641638	0.320819	+	0.947141i	$0.396042\pi$
$284$	0	0
$285$	−1253.07	−0.260441
$286$	0	0
$287$	−3878.45	−0.797692
$288$	0	0
$289$	−4892.35	−0.995797
$290$	0	0
$291$	628.166	0.126542
$292$	0	0
$293$	2847.83	0.567822	0.283911	−	0.958851i	$-0.408368\pi$
$293$	2847.83	0.567822	0.283911	+	0.958851i	$0.408368\pi$
$294$	0	0
$295$	−8533.21	−1.68414
$296$	0	0
$297$	1591.24	0.310887
$298$	0	0
$299$	995.522	0.192550
$300$	0	0
$301$	8919.53	1.70802
$302$	0	0
$303$	−740.705	−0.140437
$304$	0	0
$305$	2368.70	0.444693
$306$	0	0
$307$	2470.79	0.459334	0.229667	−	0.973269i	$-0.426236\pi$
$307$	2470.79	0.459334	0.229667	+	0.973269i	$0.426236\pi$
$308$	0	0
$309$	921.955	0.169735
$310$	0	0
$311$	5653.29	1.03077	0.515384	−	0.856959i	$-0.327649\pi$
$311$	5653.29	1.03077	0.515384	+	0.856959i	$0.327649\pi$
$312$	0	0
$313$	−8289.72	−1.49701	−0.748503	−	0.663132i	$-0.769227\pi$
$313$	−8289.72	−1.49701	−0.748503	+	0.663132i	$0.769227\pi$
$314$	0	0
$315$	6235.72	1.11537
$316$	0	0
$317$	2577.06	0.456600	0.228300	−	0.973591i	$-0.426683\pi$
$317$	2577.06	0.456600	0.228300	+	0.973591i	$0.426683\pi$
$318$	0	0
$319$	−1140.45	−0.200166
$320$	0	0
$321$	535.486	0.0931088
$322$	0	0
$323$	705.502	0.121533
$324$	0	0
$325$	−271.721	−0.0463765
$326$	0	0
$327$	−1133.11	−0.191625
$328$	0	0
$329$	−5042.70	−0.845024
$330$	0	0
$331$	−7672.12	−1.27401	−0.637006	−	0.770859i	$-0.719827\pi$
$331$	−7672.12	−1.27401	−0.637006	+	0.770859i	$0.719827\pi$
$332$	0	0
$333$	−6215.38	−1.02283
$334$	0	0
$335$	−5589.30	−0.911570
$336$	0	0
$337$	3650.77	0.590120	0.295060	−	0.955479i	$-0.404660\pi$
$337$	3650.77	0.590120	0.295060	+	0.955479i	$0.404660\pi$
$338$	0	0
$339$	−808.280	−0.129498
$340$	0	0
$341$	−2277.15	−0.361626
$342$	0	0
$343$	−4333.79	−0.682223
$344$	0	0
$345$	−338.034	−0.0527512
$346$	0	0
$347$	8737.06	1.35167	0.675835	−	0.737053i	$-0.263783\pi$
$347$	8737.06	1.35167	0.675835	+	0.737053i	$0.263783\pi$
$348$	0	0
$349$	−2143.83	−0.328816	−0.164408	−	0.986392i	$-0.552571\pi$
$349$	−2143.83	−0.328816	−0.164408	+	0.986392i	$0.552571\pi$
$350$	0	0
$351$	−961.787	−0.146258
$352$	0	0
$353$	−5111.17	−0.770651	−0.385326	−	0.922781i	$-0.625911\pi$
$353$	−5111.17	−0.770651	−0.385326	+	0.922781i	$0.625911\pi$
$354$	0	0
$355$	−3004.35	−0.449167
$356$	0	0
$357$	76.2035	0.0112972
$358$	0	0
$359$	−4520.87	−0.664630	−0.332315	−	0.943168i	$-0.607830\pi$
$359$	−4520.87	−0.664630	−0.332315	+	0.943168i	$0.607830\pi$
$360$	0	0
$361$	17245.1	2.51422
$362$	0	0
$363$	163.231	0.0236017
$364$	0	0
$365$	−13098.3	−1.87835
$366$	0	0
$367$	486.272	0.0691641	0.0345820	−	0.999402i	$-0.488990\pi$
$367$	486.272	0.0691641	0.0345820	+	0.999402i	$0.488990\pi$
$368$	0	0
$369$	4628.89	0.653036
$370$	0	0
$371$	14913.5	2.08698
$372$	0	0
$373$	10053.1	1.39552	0.697758	−	0.716333i	$-0.254181\pi$
$373$	10053.1	1.39552	0.697758	+	0.716333i	$0.254181\pi$
$374$	0	0
$375$	1101.15	0.151635
$376$	0	0
$377$	689.317	0.0941688
$378$	0	0
$379$	11602.8	1.57255	0.786273	−	0.617879i	$-0.212008\pi$
$379$	11602.8	1.57255	0.786273	+	0.617879i	$0.212008\pi$
$380$	0	0
$381$	893.363	0.120127
$382$	0	0
$383$	4161.38	0.555187	0.277593	−	0.960699i	$-0.410463\pi$
$383$	4161.38	0.555187	0.277593	+	0.960699i	$0.410463\pi$
$384$	0	0
$385$	9279.56	1.22839
$386$	0	0
$387$	−10645.4	−1.39828
$388$	0	0
$389$	−6843.95	−0.892036	−0.446018	−	0.895024i	$-0.647158\pi$
$389$	−6843.95	−0.892036	−0.446018	+	0.895024i	$0.647158\pi$
$390$	0	0
$391$	190.319	0.0246160
$392$	0	0
$393$	1785.45	0.229171
$394$	0	0
$395$	6513.97	0.829755
$396$	0	0
$397$	5474.73	0.692112	0.346056	−	0.938214i	$-0.387521\pi$
$397$	5474.73	0.692112	0.346056	+	0.938214i	$0.387521\pi$
$398$	0	0
$399$	2603.55	0.326669
$400$	0	0
$401$	890.013	0.110836	0.0554178	−	0.998463i	$-0.482351\pi$
$401$	890.013	0.110836	0.0554178	+	0.998463i	$0.482351\pi$
$402$	0	0
$403$	1376.37	0.170128
$404$	0	0
$405$	−7277.22	−0.892860
$406$	0	0
$407$	−9249.29	−1.12646
$408$	0	0
$409$	8107.81	0.980209	0.490104	−	0.871664i	$-0.336959\pi$
$409$	8107.81	0.980209	0.490104	+	0.871664i	$0.336959\pi$
$410$	0	0
$411$	547.184	0.0656706
$412$	0	0
$413$	17729.8	2.11241
$414$	0	0
$415$	5493.51	0.649797
$416$	0	0
$417$	−1058.97	−0.124360
$418$	0	0
$419$	−4237.59	−0.494080	−0.247040	−	0.969005i	$-0.579458\pi$
$419$	−4237.59	−0.494080	−0.247040	+	0.969005i	$0.579458\pi$
$420$	0	0
$421$	−953.634	−0.110397	−0.0551987	−	0.998475i	$-0.517579\pi$
$421$	−953.634	−0.110397	−0.0551987	+	0.998475i	$0.517579\pi$
$422$	0	0
$423$	6018.41	0.691785
$424$	0	0
$425$	−51.9463	−0.00592886
$426$	0	0
$427$	−4921.53	−0.557774
$428$	0	0
$429$	−707.949	−0.0796738
$430$	0	0
$431$	10098.8	1.12864	0.564318	−	0.825557i	$-0.309139\pi$
$431$	10098.8	1.12864	0.564318	+	0.825557i	$0.309139\pi$
$432$	0	0
$433$	−12833.1	−1.42430	−0.712148	−	0.702029i	$-0.752278\pi$
$433$	−12833.1	−1.42430	−0.712148	+	0.702029i	$0.752278\pi$
$434$	0	0
$435$	−234.061	−0.0257985
$436$	0	0
$437$	6502.42	0.711792
$438$	0	0
$439$	14434.8	1.56932	0.784662	−	0.619924i	$-0.212836\pi$
$439$	14434.8	1.56932	0.784662	+	0.619924i	$0.212836\pi$
$440$	0	0
$441$	−3891.93	−0.420249
$442$	0	0
$443$	7020.65	0.752959	0.376480	−	0.926425i	$-0.377134\pi$
$443$	7020.65	0.752959	0.376480	+	0.926425i	$0.377134\pi$
$444$	0	0
$445$	3817.49	0.406666
$446$	0	0
$447$	2143.80	0.226842
$448$	0	0
$449$	13210.5	1.38851	0.694254	−	0.719730i	$-0.255735\pi$
$449$	13210.5	1.38851	0.694254	+	0.719730i	$0.255735\pi$
$450$	0	0
$451$	6888.38	0.719205
$452$	0	0
$453$	−1291.74	−0.133976
$454$	0	0
$455$	−5608.79	−0.577900
$456$	0	0
$457$	−2811.18	−0.287749	−0.143875	−	0.989596i	$-0.545956\pi$
$457$	−2811.18	−0.287749	−0.143875	+	0.989596i	$0.545956\pi$
$458$	0	0
$459$	−183.870	−0.0186979
$460$	0	0
$461$	−9645.10	−0.974441	−0.487220	−	0.873279i	$-0.661989\pi$
$461$	−9645.10	−0.974441	−0.487220	+	0.873279i	$0.661989\pi$
$462$	0	0
$463$	−6923.23	−0.694924	−0.347462	−	0.937694i	$-0.612956\pi$
$463$	−6923.23	−0.694924	−0.347462	+	0.937694i	$0.612956\pi$
$464$	0	0
$465$	−467.352	−0.0466084
$466$	0	0
$467$	124.351	0.0123218	0.00616089	−	0.999981i	$-0.498039\pi$
$467$	124.351	0.0123218	0.00616089	+	0.999981i	$0.498039\pi$
$468$	0	0
$469$	11613.1	1.14337
$470$	0	0
$471$	−2022.78	−0.197887
$472$	0	0
$473$	−15841.7	−1.53996
$474$	0	0
$475$	−1774.79	−0.171438
$476$	0	0
$477$	−17799.1	−1.70852
$478$	0	0
$479$	4461.48	0.425575	0.212788	−	0.977098i	$-0.431746\pi$
$479$	4461.48	0.425575	0.212788	+	0.977098i	$0.431746\pi$
$480$	0	0
$481$	5590.50	0.529948
$482$	0	0
$483$	702.347	0.0661654
$484$	0	0
$485$	−8838.96	−0.827539
$486$	0	0
$487$	5660.72	0.526718	0.263359	−	0.964698i	$-0.415170\pi$
$487$	5660.72	0.526718	0.263359	+	0.964698i	$0.415170\pi$
$488$	0	0
$489$	1629.77	0.150718
$490$	0	0
$491$	−5587.50	−0.513565	−0.256782	−	0.966469i	$-0.582662\pi$
$491$	−5587.50	−0.513565	−0.256782	+	0.966469i	$0.582662\pi$
$492$	0	0
$493$	131.781	0.0120387
$494$	0	0
$495$	−11075.1	−1.00563
$496$	0	0
$497$	6242.24	0.563386
$498$	0	0
$499$	4210.19	0.377703	0.188851	−	0.982006i	$-0.439524\pi$
$499$	4210.19	0.377703	0.188851	+	0.982006i	$0.439524\pi$
$500$	0	0
$501$	−756.895	−0.0674962
$502$	0	0
$503$	−10796.7	−0.957063	−0.478532	−	0.878070i	$-0.658831\pi$
$503$	−10796.7	−0.957063	−0.478532	+	0.878070i	$0.658831\pi$
$504$	0	0
$505$	10422.5	0.918407
$506$	0	0
$507$	−1236.02	−0.108271
$508$	0	0
$509$	14983.6	1.30479	0.652395	−	0.757879i	$-0.273764\pi$
$509$	14983.6	1.30479	0.652395	+	0.757879i	$0.273764\pi$
$510$	0	0
$511$	27214.8	2.35599
$512$	0	0
$513$	−6282.07	−0.540663
$514$	0	0
$515$	−12972.9	−1.11001
$516$	0	0
$517$	8956.17	0.761880
$518$	0	0
$519$	−1919.38	−0.162334
$520$	0	0
$521$	−10770.3	−0.905671	−0.452836	−	0.891594i	$-0.649588\pi$
$521$	−10770.3	−0.905671	−0.452836	+	0.891594i	$0.649588\pi$
$522$	0	0
$523$	7538.93	0.630314	0.315157	−	0.949040i	$-0.397943\pi$
$523$	7538.93	0.630314	0.315157	+	0.949040i	$0.397943\pi$
$524$	0	0
$525$	−191.700	−0.0159362
$526$	0	0
$527$	263.127	0.0217495
$528$	0	0
$529$	−10412.9	−0.855829
$530$	0	0
$531$	−21160.3	−1.72934
$532$	0	0
$533$	−4163.51	−0.338352
$534$	0	0
$535$	−7534.86	−0.608898
$536$	0	0
$537$	−2688.83	−0.216073
$538$	0	0
$539$	−5791.69	−0.462831
$540$	0	0
$541$	595.816	0.0473496	0.0236748	−	0.999720i	$-0.492463\pi$
$541$	595.816	0.0473496	0.0236748	+	0.999720i	$0.492463\pi$
$542$	0	0
$543$	−2298.34	−0.181641
$544$	0	0
$545$	15944.1	1.25316
$546$	0	0
$547$	−12703.4	−0.992978	−0.496489	−	0.868043i	$-0.665378\pi$
$547$	−12703.4	−0.992978	−0.496489	+	0.868043i	$0.665378\pi$
$548$	0	0
$549$	5873.80	0.456626
$550$	0	0
$551$	4502.39	0.348109
$552$	0	0
$553$	−13534.3	−1.04075
$554$	0	0
$555$	−1898.28	−0.145185
$556$	0	0
$557$	−14973.0	−1.13901	−0.569503	−	0.821990i	$-0.692864\pi$
$557$	−14973.0	−1.13901	−0.569503	+	0.821990i	$0.692864\pi$
$558$	0	0
$559$	9575.10	0.724479
$560$	0	0
$561$	−135.342	−0.0101857
$562$	0	0
$563$	10439.3	0.781466	0.390733	−	0.920504i	$-0.372222\pi$
$563$	10439.3	0.781466	0.390733	+	0.920504i	$0.372222\pi$
$564$	0	0
$565$	11373.4	0.846869
$566$	0	0
$567$	15120.2	1.11991
$568$	0	0
$569$	−8859.72	−0.652757	−0.326379	−	0.945239i	$-0.605828\pi$
$569$	−8859.72	−0.652757	−0.326379	+	0.945239i	$0.605828\pi$
$570$	0	0
$571$	5078.17	0.372180	0.186090	−	0.982533i	$-0.440418\pi$
$571$	5078.17	0.372180	0.186090	+	0.982533i	$0.440418\pi$
$572$	0	0
$573$	−1684.37	−0.122802
$574$	0	0
$575$	−478.775	−0.0347240
$576$	0	0
$577$	20457.2	1.47599	0.737994	−	0.674808i	$-0.235773\pi$
$577$	20457.2	1.47599	0.737994	+	0.674808i	$0.235773\pi$
$578$	0	0
$579$	−479.280	−0.0344010
$580$	0	0
$581$	−11414.1	−0.815034
$582$	0	0
$583$	−26487.3	−1.88164
$584$	0	0
$585$	6694.04	0.473102
$586$	0	0
$587$	−4355.22	−0.306234	−0.153117	−	0.988208i	$-0.548931\pi$
$587$	−4355.22	−0.306234	−0.153117	+	0.988208i	$0.548931\pi$
$588$	0	0
$589$	8989.96	0.628905
$590$	0	0
$591$	1037.43	0.0722070
$592$	0	0
$593$	4342.49	0.300716	0.150358	−	0.988632i	$-0.451957\pi$
$593$	4342.49	0.300716	0.150358	+	0.988632i	$0.451957\pi$
$594$	0	0
$595$	−1072.26	−0.0738799
$596$	0	0
$597$	−1072.71	−0.0735392
$598$	0	0
$599$	12761.8	0.870506	0.435253	−	0.900308i	$-0.356659\pi$
$599$	12761.8	0.870506	0.435253	+	0.900308i	$0.356659\pi$
$600$	0	0
$601$	−2804.87	−0.190371	−0.0951857	−	0.995460i	$-0.530344\pi$
$601$	−2804.87	−0.190371	−0.0951857	+	0.995460i	$0.530344\pi$
$602$	0	0
$603$	−13860.1	−0.936032
$604$	0	0
$605$	−2296.84	−0.154346
$606$	0	0
$607$	−12033.5	−0.804654	−0.402327	−	0.915496i	$-0.631798\pi$
$607$	−12033.5	−0.804654	−0.402327	+	0.915496i	$0.631798\pi$
$608$	0	0
$609$	486.317	0.0323589
$610$	0	0
$611$	−5413.33	−0.358429
$612$	0	0
$613$	7993.67	0.526690	0.263345	−	0.964702i	$-0.415174\pi$
$613$	7993.67	0.526690	0.263345	+	0.964702i	$0.415174\pi$
$614$	0	0
$615$	1413.74	0.0926951
$616$	0	0
$617$	9922.51	0.647432	0.323716	−	0.946154i	$-0.395068\pi$
$617$	9922.51	0.647432	0.323716	+	0.946154i	$0.395068\pi$
$618$	0	0
$619$	15401.4	1.00005	0.500027	−	0.866010i	$-0.333324\pi$
$619$	15401.4	1.00005	0.500027	+	0.866010i	$0.333324\pi$
$620$	0	0
$621$	−1694.68	−0.109509
$622$	0	0
$623$	−7931.74	−0.510078
$624$	0	0
$625$	−14065.4	−0.900185
$626$	0	0
$627$	−4624.08	−0.294527
$628$	0	0
$629$	1068.77	0.0677497
$630$	0	0
$631$	3776.97	0.238287	0.119143	−	0.992877i	$-0.461985\pi$
$631$	3776.97	0.238287	0.119143	+	0.992877i	$0.461985\pi$
$632$	0	0
$633$	678.921	0.0426298
$634$	0	0
$635$	−12570.6	−0.785587
$636$	0	0
$637$	3500.64	0.217740
$638$	0	0
$639$	−7450.05	−0.461220
$640$	0	0
$641$	464.125	0.0285988	0.0142994	−	0.999898i	$-0.495448\pi$
$641$	464.125	0.0285988	0.0142994	+	0.999898i	$0.495448\pi$
$642$	0	0
$643$	−7607.61	−0.466586	−0.233293	−	0.972406i	$-0.574950\pi$
$643$	−7607.61	−0.466586	−0.233293	+	0.972406i	$0.574950\pi$
$644$	0	0
$645$	−3251.27	−0.198479
$646$	0	0
$647$	−7776.85	−0.472550	−0.236275	−	0.971686i	$-0.575927\pi$
$647$	−7776.85	−0.472550	−0.236275	+	0.971686i	$0.575927\pi$
$648$	0	0
$649$	−31489.2	−1.90456
$650$	0	0
$651$	971.033	0.0584605
$652$	0	0
$653$	−24486.1	−1.46740	−0.733702	−	0.679471i	$-0.762209\pi$
$653$	−24486.1	−1.46740	−0.733702	+	0.679471i	$0.762209\pi$
$654$	0	0
$655$	−25123.2	−1.49869
$656$	0	0
$657$	−32480.6	−1.92875
$658$	0	0
$659$	−910.966	−0.0538486	−0.0269243	−	0.999637i	$-0.508571\pi$
$659$	−910.966	−0.0538486	−0.0269243	+	0.999637i	$0.508571\pi$
$660$	0	0
$661$	5826.64	0.342859	0.171430	−	0.985196i	$-0.445161\pi$
$661$	5826.64	0.342859	0.171430	+	0.985196i	$0.445161\pi$
$662$	0	0
$663$	81.8043	0.00479188
$664$	0	0
$665$	−36634.8	−2.13629
$666$	0	0
$667$	1214.59	0.0705081
$668$	0	0
$669$	1718.40	0.0993085
$670$	0	0
$671$	8740.97	0.502893
$672$	0	0
$673$	27236.0	1.55999	0.779994	−	0.625788i	$-0.215222\pi$
$673$	27236.0	1.55999	0.779994	+	0.625788i	$0.215222\pi$
$674$	0	0
$675$	462.551	0.0263757
$676$	0	0
$677$	8989.70	0.510343	0.255172	−	0.966896i	$-0.417868\pi$
$677$	8989.70	0.510343	0.255172	+	0.966896i	$0.417868\pi$
$678$	0	0
$679$	18365.0	1.03798
$680$	0	0
$681$	1573.86	0.0885616
$682$	0	0
$683$	−6934.65	−0.388502	−0.194251	−	0.980952i	$-0.562228\pi$
$683$	−6934.65	−0.388502	−0.194251	+	0.980952i	$0.562228\pi$
$684$	0	0
$685$	−7699.46	−0.429462
$686$	0	0
$687$	−2813.63	−0.156254
$688$	0	0
$689$	16009.6	0.885221
$690$	0	0
$691$	−5234.54	−0.288178	−0.144089	−	0.989565i	$-0.546025\pi$
$691$	−5234.54	−0.288178	−0.144089	+	0.989565i	$0.546025\pi$
$692$	0	0
$693$	23011.0	1.26135
$694$	0	0
$695$	14900.9	0.813270
$696$	0	0
$697$	−795.961	−0.0432556
$698$	0	0
$699$	1909.57	0.103328
$700$	0	0
$701$	−33713.7	−1.81647	−0.908237	−	0.418457i	$-0.862571\pi$
$701$	−33713.7	−1.81647	−0.908237	+	0.418457i	$0.862571\pi$
$702$	0	0
$703$	36515.3	1.95903
$704$	0	0
$705$	1838.12	0.0981953
$706$	0	0
$707$	−21655.2	−1.15195
$708$	0	0
$709$	−12687.3	−0.672049	−0.336025	−	0.941853i	$-0.609083\pi$
$709$	−12687.3	−0.672049	−0.336025	+	0.941853i	$0.609083\pi$
$710$	0	0
$711$	16153.1	0.852021
$712$	0	0
$713$	2425.17	0.127382
$714$	0	0
$715$	9961.59	0.521038
$716$	0	0
$717$	−2984.02	−0.155426
$718$	0	0
$719$	−30604.9	−1.58744	−0.793720	−	0.608283i	$-0.791859\pi$
$719$	−30604.9	−1.58744	−0.793720	+	0.608283i	$0.791859\pi$
$720$	0	0
$721$	26954.2	1.39227
$722$	0	0
$723$	−1494.31	−0.0768661
$724$	0	0
$725$	−331.512	−0.0169821
$726$	0	0
$727$	−3727.96	−0.190182	−0.0950911	−	0.995469i	$-0.530314\pi$
$727$	−3727.96	−0.190182	−0.0950911	+	0.995469i	$0.530314\pi$
$728$	0	0
$729$	−17218.3	−0.874782
$730$	0	0
$731$	1830.52	0.0926189
$732$	0	0
$733$	1107.21	0.0557921	0.0278960	−	0.999611i	$-0.491119\pi$
$733$	1107.21	0.0557921	0.0278960	+	0.999611i	$0.491119\pi$
$734$	0	0
$735$	−1188.66	−0.0596522
$736$	0	0
$737$	−20625.6	−1.03087
$738$	0	0
$739$	29127.7	1.44991	0.724953	−	0.688798i	$-0.241861\pi$
$739$	29127.7	1.44991	0.724953	+	0.688798i	$0.241861\pi$
$740$	0	0
$741$	2794.91	0.138561
$742$	0	0
$743$	−12162.8	−0.600552	−0.300276	−	0.953852i	$-0.597079\pi$
$743$	−12162.8	−0.600552	−0.300276	+	0.953852i	$0.597079\pi$
$744$	0	0
$745$	−30165.6	−1.48347
$746$	0	0
$747$	13622.6	0.667233
$748$	0	0
$749$	15655.5	0.763736
$750$	0	0
$751$	25067.6	1.21802	0.609008	−	0.793164i	$-0.291568\pi$
$751$	25067.6	1.21802	0.609008	+	0.793164i	$0.291568\pi$
$752$	0	0
$753$	936.587	0.0453269
$754$	0	0
$755$	18176.1	0.876156
$756$	0	0
$757$	−12020.6	−0.577140	−0.288570	−	0.957459i	$-0.593180\pi$
$757$	−12020.6	−0.577140	−0.288570	+	0.957459i	$0.593180\pi$
$758$	0	0
$759$	−1247.41	−0.0596552
$760$	0	0
$761$	9255.42	0.440879	0.220439	−	0.975401i	$-0.429251\pi$
$761$	9255.42	0.440879	0.220439	+	0.975401i	$0.429251\pi$
$762$	0	0
$763$	−33127.6	−1.57182
$764$	0	0
$765$	1279.74	0.0604823
$766$	0	0
$767$	19032.9	0.896007
$768$	0	0
$769$	28690.1	1.34537	0.672687	−	0.739928i	$-0.265140\pi$
$769$	28690.1	1.34537	0.672687	+	0.739928i	$0.265140\pi$
$770$	0	0
$771$	2210.05	0.103233
$772$	0	0
$773$	−8984.36	−0.418040	−0.209020	−	0.977911i	$-0.567027\pi$
$773$	−8984.36	−0.418040	−0.209020	+	0.977911i	$0.567027\pi$
$774$	0	0
$775$	−661.933	−0.0306804
$776$	0	0
$777$	3944.13	0.182104
$778$	0	0
$779$	−27194.7	−1.25077
$780$	0	0
$781$	−11086.6	−0.507952
$782$	0	0
$783$	−1173.43	−0.0535566
$784$	0	0
$785$	28462.7	1.29411
$786$	0	0
$787$	5257.13	0.238115	0.119058	−	0.992887i	$-0.462013\pi$
$787$	5257.13	0.238115	0.119058	+	0.992887i	$0.462013\pi$
$788$	0	0
$789$	235.379	0.0106207
$790$	0	0
$791$	−23630.9	−1.06222
$792$	0	0
$793$	−5283.26	−0.236588
$794$	0	0
$795$	−5436.14	−0.242516
$796$	0	0
$797$	7623.92	0.338837	0.169419	−	0.985544i	$-0.445811\pi$
$797$	7623.92	0.338837	0.169419	+	0.985544i	$0.445811\pi$
$798$	0	0
$799$	−1034.90	−0.0458223
$800$	0	0
$801$	9466.45	0.417579
$802$	0	0
$803$	−48335.3	−2.12418
$804$	0	0
$805$	−9882.77	−0.432698
$806$	0	0
$807$	−1393.53	−0.0607862
$808$	0	0
$809$	−28238.7	−1.22722	−0.613609	−	0.789610i	$-0.710283\pi$
$809$	−28238.7	−1.22722	−0.613609	+	0.789610i	$0.710283\pi$
$810$	0	0
$811$	−29851.2	−1.29250	−0.646251	−	0.763125i	$-0.723664\pi$
$811$	−29851.2	−1.29250	−0.646251	+	0.763125i	$0.723664\pi$
$812$	0	0
$813$	−3928.69	−0.169478
$814$	0	0
$815$	−22932.7	−0.985640
$816$	0	0
$817$	62541.4	2.67815
$818$	0	0
$819$	−13908.4	−0.593407
$820$	0	0
$821$	2043.20	0.0868552	0.0434276	−	0.999057i	$-0.486172\pi$
$821$	2043.20	0.0868552	0.0434276	+	0.999057i	$0.486172\pi$
$822$	0	0
$823$	−123.242	−0.00521988	−0.00260994	−	0.999997i	$-0.500831\pi$
$823$	−123.242	−0.00521988	−0.00260994	+	0.999997i	$0.500831\pi$
$824$	0	0
$825$	340.473	0.0143682
$826$	0	0
$827$	−34335.2	−1.44371	−0.721857	−	0.692043i	$-0.756711\pi$
$827$	−34335.2	−1.44371	−0.721857	+	0.692043i	$0.756711\pi$
$828$	0	0
$829$	−14705.0	−0.616073	−0.308037	−	0.951375i	$-0.599672\pi$
$829$	−14705.0	−0.616073	−0.308037	+	0.951375i	$0.599672\pi$
$830$	0	0
$831$	6563.58	0.273993
$832$	0	0
$833$	669.237	0.0278364
$834$	0	0
$835$	10650.3	0.441401
$836$	0	0
$837$	−2342.99	−0.0967570
$838$	0	0
$839$	22204.3	0.913679	0.456840	−	0.889549i	$-0.348981\pi$
$839$	22204.3	0.913679	0.456840	+	0.889549i	$0.348981\pi$
$840$	0	0
$841$	841.000	0.0344828
$842$	0	0
$843$	5950.54	0.243117
$844$	0	0
$845$	17392.1	0.708054
$846$	0	0
$847$	4772.22	0.193595
$848$	0	0
$849$	2313.51	0.0935212
$850$	0	0
$851$	9850.54	0.396794
$852$	0	0
$853$	−40095.2	−1.60942	−0.804710	−	0.593669i	$-0.797679\pi$
$853$	−40095.2	−1.60942	−0.804710	+	0.593669i	$0.797679\pi$
$854$	0	0
$855$	43723.2	1.74889
$856$	0	0
$857$	−12883.7	−0.513536	−0.256768	−	0.966473i	$-0.582658\pi$
$857$	−12883.7	−0.513536	−0.256768	+	0.966473i	$0.582658\pi$
$858$	0	0
$859$	34336.9	1.36386	0.681932	−	0.731416i	$-0.261140\pi$
$859$	34336.9	1.36386	0.681932	+	0.731416i	$0.261140\pi$
$860$	0	0
$861$	−2937.38	−0.116267
$862$	0	0
$863$	−10864.2	−0.428531	−0.214266	−	0.976775i	$-0.568736\pi$
$863$	−10864.2	−0.428531	−0.214266	+	0.976775i	$0.568736\pi$
$864$	0	0
$865$	27007.7	1.06161
$866$	0	0
$867$	−3705.27	−0.145141
$868$	0	0
$869$	24037.8	0.938352
$870$	0	0
$871$	12466.6	0.484978
$872$	0	0
$873$	−21918.5	−0.849745
$874$	0	0
$875$	32193.1	1.24380
$876$	0	0
$877$	4818.40	0.185525	0.0927626	−	0.995688i	$-0.470430\pi$
$877$	4818.40	0.185525	0.0927626	+	0.995688i	$0.470430\pi$
$878$	0	0
$879$	2156.83	0.0827623
$880$	0	0
$881$	−30195.7	−1.15473	−0.577366	−	0.816485i	$-0.695920\pi$
$881$	−30195.7	−1.15473	−0.577366	+	0.816485i	$0.695920\pi$
$882$	0	0
$883$	−30734.0	−1.17133	−0.585663	−	0.810555i	$-0.699166\pi$
$883$	−30734.0	−1.17133	−0.585663	+	0.810555i	$0.699166\pi$
$884$	0	0
$885$	−6462.71	−0.245471
$886$	0	0
$887$	43409.3	1.64323	0.821614	−	0.570045i	$-0.193074\pi$
$887$	43409.3	1.64323	0.821614	+	0.570045i	$0.193074\pi$
$888$	0	0
$889$	26118.3	0.985355
$890$	0	0
$891$	−26854.4	−1.00972
$892$	0	0
$893$	−35358.1	−1.32499
$894$	0	0
$895$	37834.7	1.41304
$896$	0	0
$897$	753.968	0.0280650
$898$	0	0
$899$	1679.23	0.0622976
$900$	0	0
$901$	3060.64	0.113169
$902$	0	0
$903$	6755.29	0.248950
$904$	0	0
$905$	32340.1	1.18787
$906$	0	0
$907$	7991.51	0.292562	0.146281	−	0.989243i	$-0.453270\pi$
$907$	7991.51	0.292562	0.146281	+	0.989243i	$0.453270\pi$
$908$	0	0
$909$	25845.3	0.943052
$910$	0	0
$911$	−24188.5	−0.879694	−0.439847	−	0.898073i	$-0.644967\pi$
$911$	−24188.5	−0.879694	−0.439847	+	0.898073i	$0.644967\pi$
$912$	0	0
$913$	20272.1	0.734840
$914$	0	0
$915$	1793.96	0.0648157
$916$	0	0
$917$	52199.4	1.87980
$918$	0	0
$919$	−27421.3	−0.984273	−0.492136	−	0.870518i	$-0.663784\pi$
$919$	−27421.3	−0.984273	−0.492136	+	0.870518i	$0.663784\pi$
$920$	0	0
$921$	1871.28	0.0669497
$922$	0	0
$923$	6701.03	0.238968
$924$	0	0
$925$	−2688.63	−0.0955694
$926$	0	0
$927$	−32169.6	−1.13979
$928$	0	0
$929$	−9284.12	−0.327882	−0.163941	−	0.986470i	$-0.552421\pi$
$929$	−9284.12	−0.327882	−0.163941	+	0.986470i	$0.552421\pi$
$930$	0	0
$931$	22865.0	0.804910
$932$	0	0
$933$	4281.57	0.150238
$934$	0	0
$935$	1904.41	0.0666106
$936$	0	0
$937$	−1891.00	−0.0659297	−0.0329649	−	0.999457i	$-0.510495\pi$
$937$	−1891.00	−0.0659297	−0.0329649	+	0.999457i	$0.510495\pi$
$938$	0	0
$939$	−6278.30	−0.218194
$940$	0	0
$941$	15163.9	0.525322	0.262661	−	0.964888i	$-0.415400\pi$
$941$	15163.9	0.525322	0.262661	+	0.964888i	$0.415400\pi$
$942$	0	0
$943$	−7336.16	−0.253338
$944$	0	0
$945$	9547.87	0.328669
$946$	0	0
$947$	31251.8	1.07238	0.536192	−	0.844096i	$-0.319862\pi$
$947$	31251.8	1.07238	0.536192	+	0.844096i	$0.319862\pi$
$948$	0	0
$949$	29215.1	0.999327
$950$	0	0
$951$	1951.76	0.0665512
$952$	0	0
$953$	−12521.4	−0.425612	−0.212806	−	0.977094i	$-0.568260\pi$
$953$	−12521.4	−0.425612	−0.212806	+	0.977094i	$0.568260\pi$
$954$	0	0
$955$	23700.9	0.803081
$956$	0	0
$957$	−863.731	−0.0291750
$958$	0	0
$959$	15997.5	0.538670
$960$	0	0
$961$	−26438.1	−0.887451
$962$	0	0
$963$	−18684.6	−0.625237
$964$	0	0
$965$	6743.98	0.224970
$966$	0	0
$967$	18348.5	0.610183	0.305091	−	0.952323i	$-0.401313\pi$
$967$	18348.5	0.610183	0.305091	+	0.952323i	$0.401313\pi$
$968$	0	0
$969$	534.319	0.0177139
$970$	0	0
$971$	−27188.0	−0.898564	−0.449282	−	0.893390i	$-0.648320\pi$
$971$	−27188.0	−0.898564	−0.449282	+	0.893390i	$0.648320\pi$
$972$	0	0
$973$	−30960.1	−1.02008
$974$	0	0
$975$	−205.790	−0.00675955
$976$	0	0
$977$	4733.67	0.155009	0.0775044	−	0.996992i	$-0.475305\pi$
$977$	4733.67	0.155009	0.0775044	+	0.996992i	$0.475305\pi$
$978$	0	0
$979$	14087.3	0.459890
$980$	0	0
$981$	39537.5	1.28678
$982$	0	0
$983$	43971.5	1.42673	0.713363	−	0.700795i	$-0.247171\pi$
$983$	43971.5	1.42673	0.713363	+	0.700795i	$0.247171\pi$
$984$	0	0
$985$	−14597.8	−0.472208
$986$	0	0
$987$	−3819.13	−0.123166
$988$	0	0
$989$	16871.4	0.542448
$990$	0	0
$991$	14031.4	0.449769	0.224885	−	0.974385i	$-0.427799\pi$
$991$	14031.4	0.449769	0.224885	+	0.974385i	$0.427799\pi$
$992$	0	0
$993$	−5810.56	−0.185692
$994$	0	0
$995$	15094.1	0.480920
$996$	0	0
$997$	43177.4	1.37156	0.685779	−	0.727810i	$-0.259462\pi$
$997$	43177.4	1.37156	0.685779	+	0.727810i	$0.259462\pi$
$998$	0	0
$999$	−9516.73	−0.301397

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	464.4.a.f.1.1		2
4.3	odd	2		29.4.a.a.1.1	✓	2
8.3	odd	2		1856.4.a.n.1.1		2
8.5	even	2		1856.4.a.h.1.2		2
12.11	even	2		261.4.a.b.1.2		2
20.19	odd	2		725.4.a.b.1.2		2
28.27	even	2		1421.4.a.c.1.1		2
116.115	odd	2		841.4.a.a.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
29.4.a.a.1.1	✓	2	4.3	odd	2
261.4.a.b.1.2		2	12.11	even	2
464.4.a.f.1.1		2	1.1	even	1	trivial
725.4.a.b.1.2		2	20.19	odd	2
841.4.a.a.1.2		2	116.115	odd	2
1421.4.a.c.1.1		2	28.27	even	2
1856.4.a.h.1.2		2	8.5	even	2
1856.4.a.n.1.1		2	8.3	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:	\( N \)	\(=\)	\( 464 = 2^{4} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	464.a (trivial)

\(f(q)\)	\(=\)	\(q+0.757359 q^{3} -10.6569 q^{5} +22.1421 q^{7} -26.4264 q^{9} +O(q^{10})\)	\(q+0.757359 q^{3} -10.6569 q^{5} +22.1421 q^{7} -26.4264 q^{9} -39.3259 q^{11} +23.7696 q^{13} -8.07107 q^{15} +4.54416 q^{17} +155.255 q^{19} +16.7696 q^{21} +41.8823 q^{23} -11.4315 q^{25} -40.4630 q^{27} +29.0000 q^{29} +57.9045 q^{31} -29.7838 q^{33} -235.966 q^{35} +235.196 q^{37} +18.0021 q^{39} -175.161 q^{41} +402.831 q^{43} +281.622 q^{45} -227.742 q^{47} +147.274 q^{49} +3.44156 q^{51} +673.534 q^{53} +419.090 q^{55} +117.584 q^{57} +800.725 q^{59} -222.270 q^{61} -585.137 q^{63} -253.309 q^{65} +524.479 q^{67} +31.7199 q^{69} +281.917 q^{71} +1229.10 q^{73} -8.65772 q^{75} -870.759 q^{77} -611.247 q^{79} +682.868 q^{81} -515.490 q^{83} -48.4264 q^{85} +21.9634 q^{87} -358.219 q^{89} +526.309 q^{91} +43.8545 q^{93} -1654.53 q^{95} +829.415 q^{97} +1039.24 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 10 q^{3} - 10 q^{5} + 16 q^{7} + 32 q^{9}+O(q^{10}) \) 2 * q + 10 * q^3 - 10 * q^5 + 16 * q^7 + 32 * q^9	\( 2 q + 10 q^{3} - 10 q^{5} + 16 q^{7} + 32 q^{9} + 26 q^{11} - 26 q^{13} - 2 q^{15} + 60 q^{17} + 220 q^{19} - 40 q^{21} - 52 q^{23} - 136 q^{25} + 250 q^{27} + 58 q^{29} + 294 q^{31} + 574 q^{33} - 240 q^{35} + 312 q^{37} - 442 q^{39} + 40 q^{41} + 322 q^{43} + 320 q^{45} + 130 q^{47} - 158 q^{49} + 516 q^{51} + 1002 q^{53} + 462 q^{55} + 716 q^{57} + 900 q^{59} - 948 q^{61} - 944 q^{63} - 286 q^{65} - 320 q^{67} - 836 q^{69} + 660 q^{71} + 648 q^{73} - 1160 q^{75} - 1272 q^{77} - 258 q^{79} + 1790 q^{81} - 1212 q^{83} - 12 q^{85} + 290 q^{87} + 760 q^{89} + 832 q^{91} + 2226 q^{93} - 1612 q^{95} + 24 q^{97} + 4856 q^{99}+O(q^{100}) \) 2 * q + 10 * q^3 - 10 * q^5 + 16 * q^7 + 32 * q^9 + 26 * q^11 - 26 * q^13 - 2 * q^15 + 60 * q^17 + 220 * q^19 - 40 * q^21 - 52 * q^23 - 136 * q^25 + 250 * q^27 + 58 * q^29 + 294 * q^31 + 574 * q^33 - 240 * q^35 + 312 * q^37 - 442 * q^39 + 40 * q^41 + 322 * q^43 + 320 * q^45 + 130 * q^47 - 158 * q^49 + 516 * q^51 + 1002 * q^53 + 462 * q^55 + 716 * q^57 + 900 * q^59 - 948 * q^61 - 944 * q^63 - 286 * q^65 - 320 * q^67 - 836 * q^69 + 660 * q^71 + 648 * q^73 - 1160 * q^75 - 1272 * q^77 - 258 * q^79 + 1790 * q^81 - 1212 * q^83 - 12 * q^85 + 290 * q^87 + 760 * q^89 + 832 * q^91 + 2226 * q^93 - 1612 * q^95 + 24 * q^97 + 4856 * q^99

Introduction

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