LMFDB - Embedded newform 464.4.a.f.1.2

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			1.2
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)

$f(q)$	$=$	$q+9.24264 q^{3} +0.656854 q^{5} -6.14214 q^{7} +58.4264 q^{9} +O(q^{10})$	$q+9.24264 q^{3} +0.656854 q^{5} -6.14214 q^{7} +58.4264 q^{9} +65.3259 q^{11} -49.7696 q^{13} +6.07107 q^{15} +55.4558 q^{17} +64.7452 q^{19} -56.7696 q^{21} -93.8823 q^{23} -124.569 q^{25} +290.463 q^{27} +29.0000 q^{29} +236.095 q^{31} +603.784 q^{33} -4.03449 q^{35} +76.8040 q^{37} -460.002 q^{39} +215.161 q^{41} -80.8305 q^{43} +38.3776 q^{45} +357.742 q^{47} -305.274 q^{49} +512.558 q^{51} +328.466 q^{53} +42.9096 q^{55} +598.416 q^{57} +99.2750 q^{59} -725.730 q^{61} -358.863 q^{63} -32.6913 q^{65} -844.479 q^{67} -867.720 q^{69} +378.083 q^{71} -581.097 q^{73} -1151.34 q^{75} -401.241 q^{77} +353.247 q^{79} +1107.13 q^{81} -696.510 q^{83} +36.4264 q^{85} +268.037 q^{87} +1118.22 q^{89} +305.691 q^{91} +2182.15 q^{93} +42.5281 q^{95} -805.415 q^{97} +3816.76 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	9.24264	1.77875	0.889374	−	0.457181i	$-0.151141\pi$
$3$	9.24264	1.77875	0.889374	+	0.457181i	$0.151141\pi$
$4$	0	0
$5$	0.656854	0.0587508	0.0293754	−	0.999568i	$-0.490648\pi$
$5$	0.656854	0.0587508	0.0293754	+	0.999568i	$0.490648\pi$
$6$	0	0
$7$	−6.14214	−0.331644	−0.165822	−	0.986156i	$-0.553028\pi$
$7$	−6.14214	−0.331644	−0.165822	+	0.986156i	$0.553028\pi$
$8$	0	0
$9$	58.4264	2.16394
$10$	0	0
$11$	65.3259	1.79059	0.895295	−	0.445473i	$-0.146964\pi$
$11$	65.3259	1.79059	0.895295	+	0.445473i	$0.146964\pi$
$12$	0	0
$13$	−49.7696	−1.06181	−0.530907	−	0.847430i	$-0.678149\pi$
$13$	−49.7696	−1.06181	−0.530907	+	0.847430i	$0.678149\pi$
$14$	0	0
$15$	6.07107	0.104503
$16$	0	0
$17$	55.4558	0.791178	0.395589	−	0.918428i	$-0.370541\pi$
$17$	55.4558	0.791178	0.395589	+	0.918428i	$0.370541\pi$
$18$	0	0
$19$	64.7452	0.781766	0.390883	−	0.920440i	$-0.372170\pi$
$19$	64.7452	0.781766	0.390883	+	0.920440i	$0.372170\pi$
$20$	0	0
$21$	−56.7696	−0.589911
$22$	0	0
$23$	−93.8823	−0.851122	−0.425561	−	0.904930i	$-0.639923\pi$
$23$	−93.8823	−0.851122	−0.425561	+	0.904930i	$0.639923\pi$
$24$	0	0
$25$	−124.569	−0.996548
$26$	0	0
$27$	290.463	2.07036
$28$	0	0
$29$	29.0000	0.185695
$29$	29.0000	0.185695
$30$	0	0
$31$	236.095	1.36787	0.683935	−	0.729543i	$-0.260267\pi$
$31$	236.095	1.36787	0.683935	+	0.729543i	$0.260267\pi$
$32$	0	0
$33$	603.784	3.18501
$34$	0	0
$35$	−4.03449	−0.0194844
$36$	0	0
$37$	76.8040	0.341257	0.170628	−	0.985335i	$-0.445420\pi$
$37$	76.8040	0.341257	0.170628	+	0.985335i	$0.445420\pi$
$38$	0	0
$39$	−460.002	−1.88870
$40$	0	0
$41$	215.161	0.819575	0.409788	−	0.912181i	$-0.365603\pi$
$41$	215.161	0.819575	0.409788	+	0.912181i	$0.365603\pi$
$42$	0	0
$43$	−80.8305	−0.286664	−0.143332	−	0.989675i	$-0.545782\pi$
$43$	−80.8305	−0.286664	−0.143332	+	0.989675i	$0.545782\pi$
$44$	0	0
$45$	38.3776	0.127133
$46$	0	0
$47$	357.742	1.11026	0.555128	−	0.831765i	$-0.312669\pi$
$47$	357.742	1.11026	0.555128	+	0.831765i	$0.312669\pi$
$48$	0	0
$49$	−305.274	−0.890012
$50$	0	0
$51$	512.558	1.40730
$52$	0	0
$53$	328.466	0.851288	0.425644	−	0.904891i	$-0.360048\pi$
$53$	328.466	0.851288	0.425644	+	0.904891i	$0.360048\pi$
$54$	0	0
$55$	42.9096	0.105199
$56$	0	0
$57$	598.416	1.39056
$58$	0	0
$59$	99.2750	0.219059	0.109530	−	0.993984i	$-0.465066\pi$
$59$	99.2750	0.219059	0.109530	+	0.993984i	$0.465066\pi$
$60$	0	0
$61$	−725.730	−1.52328	−0.761641	−	0.647999i	$-0.775606\pi$
$61$	−725.730	−1.52328	−0.761641	+	0.647999i	$0.775606\pi$
$62$	0	0
$63$	−358.863	−0.717658
$64$	0	0
$65$	−32.6913	−0.0623825
$66$	0	0
$67$	−844.479	−1.53984	−0.769922	−	0.638138i	$-0.779705\pi$
$67$	−844.479	−1.53984	−0.769922	+	0.638138i	$0.779705\pi$
$68$	0	0
$69$	−867.720	−1.51393
$70$	0	0
$71$	378.083	0.631975	0.315988	−	0.948763i	$-0.397664\pi$
$71$	378.083	0.631975	0.315988	+	0.948763i	$0.397664\pi$
$72$	0	0
$73$	−581.097	−0.931674	−0.465837	−	0.884870i	$-0.654247\pi$
$73$	−581.097	−0.931674	−0.465837	+	0.884870i	$0.654247\pi$
$74$	0	0
$75$	−1151.34	−1.77261
$76$	0	0
$77$	−401.241	−0.593839
$78$	0	0
$79$	353.247	0.503081	0.251540	−	0.967847i	$-0.419063\pi$
$79$	353.247	0.503081	0.251540	+	0.967847i	$0.419063\pi$
$80$	0	0
$81$	1107.13	1.51870
$82$	0	0
$83$	−696.510	−0.921107	−0.460553	−	0.887632i	$-0.652349\pi$
$83$	−696.510	−0.921107	−0.460553	+	0.887632i	$0.652349\pi$
$84$	0	0
$85$	36.4264	0.0464823
$86$	0	0
$87$	268.037	0.330305
$88$	0	0
$89$	1118.22	1.33181	0.665905	−	0.746037i	$-0.268046\pi$
$89$	1118.22	1.33181	0.665905	+	0.746037i	$0.268046\pi$
$90$	0	0
$91$	305.691	0.352145
$92$	0	0
$93$	2182.15	2.43310
$94$	0	0
$95$	42.5281	0.0459294
$96$	0	0
$97$	−805.415	−0.843068	−0.421534	−	0.906813i	$-0.638508\pi$
$97$	−805.415	−0.843068	−0.421534	+	0.906813i	$0.638508\pi$
$98$	0	0
$99$	3816.76	3.87473
$100$	0	0
$101$	−1373.99	−1.35363	−0.676817	−	0.736151i	$-0.736641\pi$
$101$	−1373.99	−1.35363	−0.676817	+	0.736151i	$0.736641\pi$
$102$	0	0
$103$	634.672	0.607147	0.303573	−	0.952808i	$-0.401820\pi$
$103$	634.672	0.607147	0.303573	+	0.952808i	$0.401820\pi$
$104$	0	0
$105$	−37.2893	−0.0346578
$106$	0	0
$107$	180.956	0.163493	0.0817463	−	0.996653i	$-0.473950\pi$
$107$	180.956	0.163493	0.0817463	+	0.996653i	$0.473950\pi$
$108$	0	0
$109$	1038.14	0.912251	0.456125	−	0.889916i	$-0.349237\pi$
$109$	1038.14	0.912251	0.456125	+	0.889916i	$0.349237\pi$
$110$	0	0
$111$	709.872	0.607010
$112$	0	0
$113$	−184.765	−0.153816	−0.0769082	−	0.997038i	$-0.524505\pi$
$113$	−184.765	−0.153816	−0.0769082	+	0.997038i	$0.524505\pi$
$114$	0	0
$115$	−61.6670	−0.0500041
$116$	0	0
$117$	−2907.86	−2.29770
$118$	0	0
$119$	−340.617	−0.262389
$120$	0	0
$121$	2936.47	2.20622
$122$	0	0
$123$	1988.66	1.45782
$124$	0	0
$125$	−163.930	−0.117299
$126$	0	0
$127$	−1999.58	−1.39712	−0.698558	−	0.715554i	$-0.746175\pi$
$127$	−1999.58	−1.39712	−0.698558	+	0.715554i	$0.746175\pi$
$128$	0	0
$129$	−747.087	−0.509902
$130$	0	0
$131$	−561.468	−0.374471	−0.187236	−	0.982315i	$-0.559953\pi$
$131$	−561.468	−0.374471	−0.187236	+	0.982315i	$0.559953\pi$
$132$	0	0
$133$	−397.674	−0.259268
$134$	0	0
$135$	190.792	0.121635
$136$	0	0
$137$	−250.489	−0.156210	−0.0781050	−	0.996945i	$-0.524887\pi$
$137$	−250.489	−0.156210	−0.0781050	+	0.996945i	$0.524887\pi$
$138$	0	0
$139$	242.244	0.147819	0.0739096	−	0.997265i	$-0.476452\pi$
$139$	242.244	0.147819	0.0739096	+	0.997265i	$0.476452\pi$
$140$	0	0
$141$	3306.48	1.97487
$142$	0	0
$143$	−3251.24	−1.90128
$144$	0	0
$145$	19.0488	0.0109098
$146$	0	0
$147$	−2821.54	−1.58311
$148$	0	0
$149$	−1632.63	−0.897652	−0.448826	−	0.893619i	$-0.648158\pi$
$149$	−1632.63	−0.897652	−0.448826	+	0.893619i	$0.648158\pi$
$150$	0	0
$151$	121.582	0.0655245	0.0327623	−	0.999463i	$-0.489570\pi$
$151$	121.582	0.0655245	0.0327623	+	0.999463i	$0.489570\pi$
$152$	0	0
$153$	3240.09	1.71206
$154$	0	0
$155$	155.080	0.0803635
$156$	0	0
$157$	−753.163	−0.382860	−0.191430	−	0.981506i	$-0.561312\pi$
$157$	−753.163	−0.382860	−0.191430	+	0.981506i	$0.561312\pi$
$158$	0	0
$159$	3035.89	1.51423
$160$	0	0
$161$	576.638	0.282270
$162$	0	0
$163$	−537.917	−0.258484	−0.129242	−	0.991613i	$-0.541254\pi$
$163$	−537.917	−0.258484	−0.129242	+	0.991613i	$0.541254\pi$
$164$	0	0
$165$	396.598	0.187122
$166$	0	0
$167$	−484.613	−0.224554	−0.112277	−	0.993677i	$-0.535814\pi$
$167$	−484.613	−0.224554	−0.112277	+	0.993677i	$0.535814\pi$
$168$	0	0
$169$	280.008	0.127450
$170$	0	0
$171$	3782.83	1.69170
$172$	0	0
$173$	−3269.70	−1.43694	−0.718469	−	0.695559i	$-0.755157\pi$
$173$	−3269.70	−1.43694	−0.718469	+	0.695559i	$0.755157\pi$
$174$	0	0
$175$	765.117	0.330499
$176$	0	0
$177$	917.563	0.389651
$178$	0	0
$179$	562.267	0.234781	0.117390	−	0.993086i	$-0.462547\pi$
$179$	562.267	0.234781	0.117390	+	0.993086i	$0.462547\pi$
$180$	0	0
$181$	−1507.32	−0.618998	−0.309499	−	0.950900i	$-0.600161\pi$
$181$	−1507.32	−0.618998	−0.309499	+	0.950900i	$0.600161\pi$
$182$	0	0
$183$	−6707.66	−2.70953
$184$	0	0
$185$	50.4491	0.0200491
$186$	0	0
$187$	3622.70	1.41668
$188$	0	0
$189$	−1784.06	−0.686622
$190$	0	0
$191$	−4532.00	−1.71688	−0.858439	−	0.512915i	$-0.828566\pi$
$191$	−4532.00	−1.71688	−0.858439	+	0.512915i	$0.828566\pi$
$192$	0	0
$193$	−2935.17	−1.09471	−0.547353	−	0.836902i	$-0.684364\pi$
$193$	−2935.17	−1.09471	−0.547353	+	0.836902i	$0.684364\pi$
$194$	0	0
$195$	−302.154	−0.110963
$196$	0	0
$197$	2682.20	0.970043	0.485022	−	0.874502i	$-0.338812\pi$
$197$	2682.20	0.970043	0.485022	+	0.874502i	$0.338812\pi$
$198$	0	0
$199$	648.376	0.230966	0.115483	−	0.993309i	$-0.463158\pi$
$199$	648.376	0.230966	0.115483	+	0.993309i	$0.463158\pi$
$200$	0	0
$201$	−7805.22	−2.73899
$202$	0	0
$203$	−178.122	−0.0615848
$204$	0	0
$205$	141.330	0.0481507
$206$	0	0
$207$	−5485.20	−1.84178
$208$	0	0
$209$	4229.54	1.39982
$210$	0	0
$211$	4949.57	1.61489	0.807446	−	0.589941i	$-0.200849\pi$
$211$	4949.57	1.61489	0.807446	+	0.589941i	$0.200849\pi$
$212$	0	0
$213$	3494.49	1.12412
$214$	0	0
$215$	−53.0939	−0.0168417
$216$	0	0
$217$	−1450.13	−0.453646
$218$	0	0
$219$	−5370.87	−1.65721
$220$	0	0
$221$	−2760.01	−0.840084
$222$	0	0
$223$	−2216.94	−0.665729	−0.332864	−	0.942975i	$-0.608015\pi$
$223$	−2216.94	−0.665729	−0.332864	+	0.942975i	$0.608015\pi$
$224$	0	0
$225$	−7278.09	−2.15647
$226$	0	0
$227$	−4546.09	−1.32923	−0.664613	−	0.747187i	$-0.731404\pi$
$227$	−4546.09	−1.32923	−0.664613	+	0.747187i	$0.731404\pi$
$228$	0	0
$229$	3339.05	0.963539	0.481770	−	0.876298i	$-0.339994\pi$
$229$	3339.05	0.963539	0.481770	+	0.876298i	$0.339994\pi$
$230$	0	0
$231$	−3708.52	−1.05629
$232$	0	0
$233$	−3995.35	−1.12336	−0.561682	−	0.827353i	$-0.689846\pi$
$233$	−3995.35	−1.12336	−0.561682	+	0.827353i	$0.689846\pi$
$234$	0	0
$235$	234.984	0.0652285
$236$	0	0
$237$	3264.93	0.894853
$238$	0	0
$239$	1400.04	0.378915	0.189458	−	0.981889i	$-0.439327\pi$
$239$	1400.04	0.378915	0.189458	+	0.981889i	$0.439327\pi$
$240$	0	0
$241$	−2040.94	−0.545513	−0.272756	−	0.962083i	$-0.587935\pi$
$241$	−2040.94	−0.545513	−0.272756	+	0.962083i	$0.587935\pi$
$242$	0	0
$243$	2390.32	0.631026
$244$	0	0
$245$	−200.521	−0.0522890
$246$	0	0
$247$	−3222.34	−0.830091
$248$	0	0
$249$	−6437.59	−1.63842
$250$	0	0
$251$	−802.648	−0.201843	−0.100922	−	0.994894i	$-0.532179\pi$
$251$	−802.648	−0.201843	−0.100922	+	0.994894i	$0.532179\pi$
$252$	0	0
$253$	−6132.94	−1.52401
$254$	0	0
$255$	336.676	0.0826803
$256$	0	0
$257$	−4464.10	−1.08351	−0.541756	−	0.840536i	$-0.682240\pi$
$257$	−4464.10	−1.08351	−0.541756	+	0.840536i	$0.682240\pi$
$258$	0	0
$259$	−471.741	−0.113176
$260$	0	0
$261$	1694.37	0.401834
$262$	0	0
$263$	3815.21	0.894509	0.447255	−	0.894407i	$-0.352402\pi$
$263$	3815.21	0.894509	0.447255	+	0.894407i	$0.352402\pi$
$264$	0	0
$265$	215.754	0.0500139
$266$	0	0
$267$	10335.3	2.36895
$268$	0	0
$269$	4523.98	1.02540	0.512699	−	0.858569i	$-0.328646\pi$
$269$	4523.98	1.02540	0.512699	+	0.858569i	$0.328646\pi$
$270$	0	0
$271$	−3962.65	−0.888242	−0.444121	−	0.895967i	$-0.646484\pi$
$271$	−3962.65	−0.888242	−0.444121	+	0.895967i	$0.646484\pi$
$272$	0	0
$273$	2825.40	0.626376
$274$	0	0
$275$	−8137.55	−1.78441
$276$	0	0
$277$	2217.59	0.481019	0.240509	−	0.970647i	$-0.422685\pi$
$277$	2217.59	0.481019	0.240509	+	0.970647i	$0.422685\pi$
$278$	0	0
$279$	13794.2	2.95999
$280$	0	0
$281$	−2562.96	−0.544105	−0.272053	−	0.962282i	$-0.587702\pi$
$281$	−2562.96	−0.544105	−0.272053	+	0.962282i	$0.587702\pi$
$282$	0	0
$283$	3869.29	0.812741	0.406370	−	0.913708i	$-0.366794\pi$
$283$	3869.29	0.812741	0.406370	+	0.913708i	$0.366794\pi$
$284$	0	0
$285$	393.072	0.0816968
$286$	0	0
$287$	−1321.55	−0.271807
$288$	0	0
$289$	−1837.65	−0.374038
$290$	0	0
$291$	−7444.17	−1.49960
$292$	0	0
$293$	−3883.83	−0.774388	−0.387194	−	0.921998i	$-0.626556\pi$
$293$	−3883.83	−0.774388	−0.387194	+	0.921998i	$0.626556\pi$
$294$	0	0
$295$	65.2092	0.0128699
$296$	0	0
$297$	18974.8	3.70716
$298$	0	0
$299$	4672.48	0.903734
$300$	0	0
$301$	496.472	0.0950703
$302$	0	0
$303$	−12699.3	−2.40777
$304$	0	0
$305$	−476.699	−0.0894941
$306$	0	0
$307$	403.210	0.0749590	0.0374795	−	0.999297i	$-0.488067\pi$
$307$	403.210	0.0749590	0.0374795	+	0.999297i	$0.488067\pi$
$308$	0	0
$309$	5866.05	1.07996
$310$	0	0
$311$	4838.71	0.882244	0.441122	−	0.897447i	$-0.354581\pi$
$311$	4838.71	0.882244	0.441122	+	0.897447i	$0.354581\pi$
$312$	0	0
$313$	−8544.28	−1.54298	−0.771488	−	0.636244i	$-0.780487\pi$
$313$	−8544.28	−1.54298	−0.771488	+	0.636244i	$0.780487\pi$
$314$	0	0
$315$	−235.721	−0.0421630
$316$	0	0
$317$	−1773.06	−0.314148	−0.157074	−	0.987587i	$-0.550206\pi$
$317$	−1773.06	−0.314148	−0.157074	+	0.987587i	$0.550206\pi$
$318$	0	0
$319$	1894.45	0.332504
$320$	0	0
$321$	1672.51	0.290812
$322$	0	0
$323$	3590.50	0.618516
$324$	0	0
$325$	6199.72	1.05815
$326$	0	0
$327$	9595.11	1.62266
$328$	0	0
$329$	−2197.30	−0.368210
$330$	0	0
$331$	−801.875	−0.133157	−0.0665786	−	0.997781i	$-0.521208\pi$
$331$	−801.875	−0.133157	−0.0665786	+	0.997781i	$0.521208\pi$
$332$	0	0
$333$	4487.38	0.738460
$334$	0	0
$335$	−554.700	−0.0904671
$336$	0	0
$337$	8193.23	1.32437	0.662186	−	0.749339i	$-0.269629\pi$
$337$	8193.23	1.32437	0.662186	+	0.749339i	$0.269629\pi$
$338$	0	0
$339$	−1707.72	−0.273601
$340$	0	0
$341$	15423.1	2.44930
$342$	0	0
$343$	3981.79	0.626811
$344$	0	0
$345$	−569.966	−0.0889447
$346$	0	0
$347$	10914.9	1.68860	0.844301	−	0.535869i	$-0.180016\pi$
$347$	10914.9	1.68860	0.844301	+	0.535869i	$0.180016\pi$
$348$	0	0
$349$	6697.83	1.02730	0.513649	−	0.858001i	$-0.328294\pi$
$349$	6697.83	1.02730	0.513649	+	0.858001i	$0.328294\pi$
$350$	0	0
$351$	−14456.2	−2.19833
$352$	0	0
$353$	−3764.83	−0.567654	−0.283827	−	0.958875i	$-0.591604\pi$
$353$	−3764.83	−0.567654	−0.283827	+	0.958875i	$0.591604\pi$
$354$	0	0
$355$	248.346	0.0371291
$356$	0	0
$357$	−3148.20	−0.466724
$358$	0	0
$359$	−6577.13	−0.966930	−0.483465	−	0.875364i	$-0.660622\pi$
$359$	−6577.13	−0.966930	−0.483465	+	0.875364i	$0.660622\pi$
$360$	0	0
$361$	−2667.06	−0.388841
$362$	0	0
$363$	27140.8	3.92430
$364$	0	0
$365$	−381.696	−0.0547366
$366$	0	0
$367$	−2274.27	−0.323477	−0.161738	−	0.986834i	$-0.551710\pi$
$367$	−2274.27	−0.323477	−0.161738	+	0.986834i	$0.551710\pi$
$368$	0	0
$369$	12571.1	1.77351
$370$	0	0
$371$	−2017.48	−0.282325
$372$	0	0
$373$	1284.94	0.178369	0.0891844	−	0.996015i	$-0.471574\pi$
$373$	1284.94	0.178369	0.0891844	+	0.996015i	$0.471574\pi$
$374$	0	0
$375$	−1515.15	−0.208645
$376$	0	0
$377$	−1443.32	−0.197174
$378$	0	0
$379$	−174.785	−0.0236890	−0.0118445	−	0.999930i	$-0.503770\pi$
$379$	−174.785	−0.0236890	−0.0118445	+	0.999930i	$0.503770\pi$
$380$	0	0
$381$	−18481.4	−2.48512
$382$	0	0
$383$	5558.62	0.741599	0.370799	−	0.928713i	$-0.379084\pi$
$383$	5558.62	0.741599	0.370799	+	0.928713i	$0.379084\pi$
$384$	0	0
$385$	−263.557	−0.0348885
$386$	0	0
$387$	−4722.64	−0.620323
$388$	0	0
$389$	−2556.05	−0.333154	−0.166577	−	0.986028i	$-0.553271\pi$
$389$	−2556.05	−0.333154	−0.166577	+	0.986028i	$0.553271\pi$
$390$	0	0
$391$	−5206.32	−0.673388
$392$	0	0
$393$	−5189.45	−0.666089
$394$	0	0
$395$	232.032	0.0295564
$396$	0	0
$397$	5927.27	0.749323	0.374662	−	0.927162i	$-0.377759\pi$
$397$	5927.27	0.749323	0.374662	+	0.927162i	$0.377759\pi$
$398$	0	0
$399$	−3675.55	−0.461173
$400$	0	0
$401$	4747.99	0.591280	0.295640	−	0.955299i	$-0.404467\pi$
$401$	4747.99	0.591280	0.295640	+	0.955299i	$0.404467\pi$
$402$	0	0
$403$	−11750.4	−1.45243
$404$	0	0
$405$	727.224	0.0892249
$406$	0	0
$407$	5017.29	0.611052
$408$	0	0
$409$	5200.19	0.628686	0.314343	−	0.949309i	$-0.398216\pi$
$409$	5200.19	0.628686	0.314343	+	0.949309i	$0.398216\pi$
$410$	0	0
$411$	−2315.18	−0.277858
$412$	0	0
$413$	−609.761	−0.0726498
$414$	0	0
$415$	−457.505	−0.0541158
$416$	0	0
$417$	2238.97	0.262933
$418$	0	0
$419$	6425.59	0.749189	0.374595	−	0.927189i	$-0.377782\pi$
$419$	6425.59	0.749189	0.374595	+	0.927189i	$0.377782\pi$
$420$	0	0
$421$	10037.6	1.16201	0.581003	−	0.813902i	$-0.302661\pi$
$421$	10037.6	1.16201	0.581003	+	0.813902i	$0.302661\pi$
$422$	0	0
$423$	20901.6	2.40253
$424$	0	0
$425$	−6908.05	−0.788447
$426$	0	0
$427$	4457.53	0.505188
$428$	0	0
$429$	−30050.1	−3.38189
$430$	0	0
$431$	−16646.8	−1.86044	−0.930218	−	0.367006i	$-0.880383\pi$
$431$	−16646.8	−1.86044	−0.930218	+	0.367006i	$0.880383\pi$
$432$	0	0
$433$	15089.1	1.67468	0.837340	−	0.546682i	$-0.184109\pi$
$433$	15089.1	1.67468	0.837340	+	0.546682i	$0.184109\pi$
$434$	0	0
$435$	176.061	0.0194057
$436$	0	0
$437$	−6078.42	−0.665378
$438$	0	0
$439$	3777.24	0.410656	0.205328	−	0.978693i	$-0.434174\pi$
$439$	3777.24	0.410656	0.205328	+	0.978693i	$0.434174\pi$
$440$	0	0
$441$	−17836.1	−1.92593
$442$	0	0
$443$	−7992.65	−0.857206	−0.428603	−	0.903493i	$-0.640994\pi$
$443$	−7992.65	−0.857206	−0.428603	+	0.903493i	$0.640994\pi$
$444$	0	0
$445$	734.507	0.0782449
$446$	0	0
$447$	−15089.8	−1.59670
$448$	0	0
$449$	6433.54	0.676209	0.338104	−	0.941109i	$-0.390214\pi$
$449$	6433.54	0.676209	0.338104	+	0.941109i	$0.390214\pi$
$450$	0	0
$451$	14055.6	1.46752
$452$	0	0
$453$	1123.74	0.116552
$454$	0	0
$455$	200.795	0.0206888
$456$	0	0
$457$	6975.18	0.713972	0.356986	−	0.934110i	$-0.383804\pi$
$457$	6975.18	0.713972	0.356986	+	0.934110i	$0.383804\pi$
$458$	0	0
$459$	16107.9	1.63802
$460$	0	0
$461$	−14758.9	−1.49109	−0.745543	−	0.666458i	$-0.767810\pi$
$461$	−14758.9	−1.49109	−0.745543	+	0.666458i	$0.767810\pi$
$462$	0	0
$463$	18951.2	1.90224	0.951121	−	0.308818i	$-0.0999334\pi$
$463$	18951.2	1.90224	0.951121	+	0.308818i	$0.0999334\pi$
$464$	0	0
$465$	1433.35	0.142946
$466$	0	0
$467$	−12442.4	−1.23290	−0.616449	−	0.787395i	$-0.711429\pi$
$467$	−12442.4	−1.23290	−0.616449	+	0.787395i	$0.711429\pi$
$468$	0	0
$469$	5186.91	0.510680
$470$	0	0
$471$	−6961.22	−0.681010
$472$	0	0
$473$	−5280.33	−0.513297
$474$	0	0
$475$	−8065.21	−0.779068
$476$	0	0
$477$	19191.1	1.84214
$478$	0	0
$479$	−12947.5	−1.23504	−0.617522	−	0.786554i	$-0.711863\pi$
$479$	−12947.5	−1.23504	−0.617522	+	0.786554i	$0.711863\pi$
$480$	0	0
$481$	−3822.50	−0.362352
$482$	0	0
$483$	5329.65	0.502086
$484$	0	0
$485$	−529.041	−0.0495309
$486$	0	0
$487$	−9844.72	−0.916030	−0.458015	−	0.888944i	$-0.651439\pi$
$487$	−9844.72	−0.916030	−0.458015	+	0.888944i	$0.651439\pi$
$488$	0	0
$489$	−4971.77	−0.459778
$490$	0	0
$491$	6809.50	0.625883	0.312941	−	0.949772i	$-0.398686\pi$
$491$	6809.50	0.625883	0.312941	+	0.949772i	$0.398686\pi$
$492$	0	0
$493$	1608.22	0.146918
$494$	0	0
$495$	2507.05	0.227644
$496$	0	0
$497$	−2322.24	−0.209591
$498$	0	0
$499$	15953.8	1.43124	0.715622	−	0.698488i	$-0.246143\pi$
$499$	15953.8	1.43124	0.715622	+	0.698488i	$0.246143\pi$
$500$	0	0
$501$	−4479.11	−0.399424
$502$	0	0
$503$	14582.7	1.29267	0.646334	−	0.763054i	$-0.276301\pi$
$503$	14582.7	1.29267	0.646334	+	0.763054i	$0.276301\pi$
$504$	0	0
$505$	−902.511	−0.0795272
$506$	0	0
$507$	2588.02	0.226702
$508$	0	0
$509$	20906.4	1.82055	0.910273	−	0.414008i	$-0.135871\pi$
$509$	20906.4	1.82055	0.910273	+	0.414008i	$0.135871\pi$
$510$	0	0
$511$	3569.17	0.308984
$512$	0	0
$513$	18806.1	1.61854
$514$	0	0
$515$	416.887	0.0356704
$516$	0	0
$517$	23369.8	1.98802
$518$	0	0
$519$	−30220.6	−2.55595
$520$	0	0
$521$	−15131.7	−1.27242	−0.636212	−	0.771515i	$-0.719500\pi$
$521$	−15131.7	−1.27242	−0.636212	+	0.771515i	$0.719500\pi$
$522$	0	0
$523$	−12146.9	−1.01558	−0.507790	−	0.861481i	$-0.669537\pi$
$523$	−12146.9	−1.01558	−0.507790	+	0.861481i	$0.669537\pi$
$524$	0	0
$525$	7071.70	0.587875
$526$	0	0
$527$	13092.9	1.08223
$528$	0	0
$529$	−3353.12	−0.275592
$530$	0	0
$531$	5800.28	0.474032
$532$	0	0
$533$	−10708.5	−0.870237
$534$	0	0
$535$	118.862	0.00960532
$536$	0	0
$537$	5196.83	0.417616
$538$	0	0
$539$	−19942.3	−1.59365
$540$	0	0
$541$	−22291.8	−1.77153	−0.885767	−	0.464130i	$-0.846367\pi$
$541$	−22291.8	−1.77153	−0.885767	+	0.464130i	$0.846367\pi$
$542$	0	0
$543$	−13931.7	−1.10104
$544$	0	0
$545$	681.904	0.0535955
$546$	0	0
$547$	15439.4	1.20684	0.603421	−	0.797423i	$-0.293804\pi$
$547$	15439.4	1.20684	0.603421	+	0.797423i	$0.293804\pi$
$548$	0	0
$549$	−42401.8	−3.29629
$550$	0	0
$551$	1877.61	0.145170
$552$	0	0
$553$	−2169.69	−0.166844
$554$	0	0
$555$	466.283	0.0356623
$556$	0	0
$557$	2336.99	0.177776	0.0888881	−	0.996042i	$-0.471669\pi$
$557$	2336.99	0.177776	0.0888881	+	0.996042i	$0.471669\pi$
$558$	0	0
$559$	4022.90	0.304384
$560$	0	0
$561$	33483.3	2.51991
$562$	0	0
$563$	−19833.3	−1.48468	−0.742340	−	0.670023i	$-0.766284\pi$
$563$	−19833.3	−1.48468	−0.742340	+	0.670023i	$0.766284\pi$
$564$	0	0
$565$	−121.364	−0.00903685
$566$	0	0
$567$	−6800.16	−0.503668
$568$	0	0
$569$	11063.7	0.815141	0.407571	−	0.913174i	$-0.366376\pi$
$569$	11063.7	0.815141	0.407571	+	0.913174i	$0.366376\pi$
$570$	0	0
$571$	665.827	0.0487986	0.0243993	−	0.999702i	$-0.492233\pi$
$571$	665.827	0.0487986	0.0243993	+	0.999702i	$0.492233\pi$
$572$	0	0
$573$	−41887.6	−3.05389
$574$	0	0
$575$	11694.8	0.848184
$576$	0	0
$577$	−7165.21	−0.516970	−0.258485	−	0.966015i	$-0.583223\pi$
$577$	−7165.21	−0.516970	−0.258485	+	0.966015i	$0.583223\pi$
$578$	0	0
$579$	−27128.7	−1.94720
$580$	0	0
$581$	4278.06	0.305480
$582$	0	0
$583$	21457.3	1.52431
$584$	0	0
$585$	−1910.04	−0.134992
$586$	0	0
$587$	10375.2	0.729525	0.364763	−	0.931101i	$-0.381150\pi$
$587$	10375.2	0.729525	0.364763	+	0.931101i	$0.381150\pi$
$588$	0	0
$589$	15286.0	1.06936
$590$	0	0
$591$	24790.6	1.72546
$592$	0	0
$593$	18931.5	1.31100	0.655501	−	0.755194i	$-0.272458\pi$
$593$	18931.5	1.31100	0.655501	+	0.755194i	$0.272458\pi$
$594$	0	0
$595$	−223.736	−0.0154156
$596$	0	0
$597$	5992.71	0.410829
$598$	0	0
$599$	12244.2	0.835199	0.417600	−	0.908631i	$-0.362871\pi$
$599$	12244.2	0.835199	0.417600	+	0.908631i	$0.362871\pi$
$600$	0	0
$601$	15596.9	1.05859	0.529293	−	0.848439i	$-0.322457\pi$
$601$	15596.9	1.05859	0.529293	+	0.848439i	$0.322457\pi$
$602$	0	0
$603$	−49339.9	−3.33213
$604$	0	0
$605$	1928.84	0.129617
$606$	0	0
$607$	10155.5	0.679076	0.339538	−	0.940592i	$-0.389729\pi$
$607$	10155.5	0.679076	0.339538	+	0.940592i	$0.389729\pi$
$608$	0	0
$609$	−1646.32	−0.109544
$610$	0	0
$611$	−17804.7	−1.17889
$612$	0	0
$613$	−6227.67	−0.410331	−0.205166	−	0.978727i	$-0.565773\pi$
$613$	−6227.67	−0.410331	−0.205166	+	0.978727i	$0.565773\pi$
$614$	0	0
$615$	1306.26	0.0856479
$616$	0	0
$617$	14357.5	0.936808	0.468404	−	0.883514i	$-0.344829\pi$
$617$	14357.5	0.936808	0.468404	+	0.883514i	$0.344829\pi$
$618$	0	0
$619$	13220.6	0.858453	0.429227	−	0.903197i	$-0.358786\pi$
$619$	13220.6	0.858453	0.429227	+	0.903197i	$0.358786\pi$
$620$	0	0
$621$	−27269.3	−1.76213
$622$	0	0
$623$	−6868.26	−0.441687
$624$	0	0
$625$	15463.4	0.989657
$626$	0	0
$627$	39092.1	2.48993
$628$	0	0
$629$	4259.23	0.269995
$630$	0	0
$631$	−1828.97	−0.115389	−0.0576943	−	0.998334i	$-0.518375\pi$
$631$	−1828.97	−0.115389	−0.0576943	+	0.998334i	$0.518375\pi$
$632$	0	0
$633$	45747.1	2.87249
$634$	0	0
$635$	−1313.43	−0.0820817
$636$	0	0
$637$	15193.4	0.945028
$638$	0	0
$639$	22090.0	1.36756
$640$	0	0
$641$	−22644.1	−1.39530	−0.697651	−	0.716437i	$-0.745772\pi$
$641$	−22644.1	−1.39530	−0.697651	+	0.716437i	$0.745772\pi$
$642$	0	0
$643$	−22728.4	−1.39397	−0.696983	−	0.717088i	$-0.745475\pi$
$643$	−22728.4	−1.39397	−0.696983	+	0.717088i	$0.745475\pi$
$644$	0	0
$645$	−490.728	−0.0299572
$646$	0	0
$647$	5844.85	0.355154	0.177577	−	0.984107i	$-0.443174\pi$
$647$	5844.85	0.355154	0.177577	+	0.984107i	$0.443174\pi$
$648$	0	0
$649$	6485.23	0.392246
$650$	0	0
$651$	−13403.0	−0.806922
$652$	0	0
$653$	15174.1	0.909355	0.454677	−	0.890656i	$-0.349755\pi$
$653$	15174.1	0.909355	0.454677	+	0.890656i	$0.349755\pi$
$654$	0	0
$655$	−368.803	−0.0220005
$656$	0	0
$657$	−33951.4	−2.01609
$658$	0	0
$659$	27857.0	1.64667	0.823333	−	0.567558i	$-0.192112\pi$
$659$	27857.0	1.64667	0.823333	+	0.567558i	$0.192112\pi$
$660$	0	0
$661$	−4966.64	−0.292254	−0.146127	−	0.989266i	$-0.546681\pi$
$661$	−4966.64	−0.292254	−0.146127	+	0.989266i	$0.546681\pi$
$662$	0	0
$663$	−25509.8	−1.49430
$664$	0	0
$665$	−261.214	−0.0152322
$666$	0	0
$667$	−2722.59	−0.158049
$668$	0	0
$669$	−20490.4	−1.18416
$670$	0	0
$671$	−47409.0	−2.72758
$672$	0	0
$673$	−2338.02	−0.133914	−0.0669569	−	0.997756i	$-0.521329\pi$
$673$	−2338.02	−0.133914	−0.0669569	+	0.997756i	$0.521329\pi$
$674$	0	0
$675$	−36182.6	−2.06321
$676$	0	0
$677$	6342.30	0.360051	0.180025	−	0.983662i	$-0.442382\pi$
$677$	6342.30	0.360051	0.180025	+	0.983662i	$0.442382\pi$
$678$	0	0
$679$	4946.97	0.279598
$680$	0	0
$681$	−42017.9	−2.36436
$682$	0	0
$683$	30366.6	1.70124	0.850620	−	0.525780i	$-0.176227\pi$
$683$	30366.6	1.70124	0.850620	+	0.525780i	$0.176227\pi$
$684$	0	0
$685$	−164.535	−0.00917746
$686$	0	0
$687$	30861.6	1.71389
$688$	0	0
$689$	−16347.6	−0.903910
$690$	0	0
$691$	11826.5	0.651089	0.325545	−	0.945527i	$-0.394452\pi$
$691$	11826.5	0.651089	0.325545	+	0.945527i	$0.394452\pi$
$692$	0	0
$693$	−23443.0	−1.28503
$694$	0	0
$695$	159.119	0.00868450
$696$	0	0
$697$	11932.0	0.648429
$698$	0	0
$699$	−36927.6	−1.99818
$700$	0	0
$701$	−2776.33	−0.149587	−0.0747936	−	0.997199i	$-0.523830\pi$
$701$	−2776.33	−0.149587	−0.0747936	+	0.997199i	$0.523830\pi$
$702$	0	0
$703$	4972.69	0.266783
$704$	0	0
$705$	2171.88	0.116025
$706$	0	0
$707$	8439.23	0.448925
$708$	0	0
$709$	−15962.7	−0.845543	−0.422772	−	0.906236i	$-0.638943\pi$
$709$	−15962.7	−0.845543	−0.422772	+	0.906236i	$0.638943\pi$
$710$	0	0
$711$	20638.9	1.08864
$712$	0	0
$713$	−22165.2	−1.16422
$714$	0	0
$715$	−2135.59	−0.111702
$716$	0	0
$717$	12940.0	0.673994
$718$	0	0
$719$	20832.9	1.08058	0.540289	−	0.841480i	$-0.318315\pi$
$719$	20832.9	1.08058	0.540289	+	0.841480i	$0.318315\pi$
$720$	0	0
$721$	−3898.24	−0.201357
$722$	0	0
$723$	−18863.7	−0.970329
$724$	0	0
$725$	−3612.49	−0.185054
$726$	0	0
$727$	−4452.04	−0.227121	−0.113561	−	0.993531i	$-0.536226\pi$
$727$	−4452.04	−0.227121	−0.113561	+	0.993531i	$0.536226\pi$
$728$	0	0
$729$	−7799.67	−0.396264
$730$	0	0
$731$	−4482.52	−0.226802
$732$	0	0
$733$	−12107.2	−0.610082	−0.305041	−	0.952339i	$-0.598670\pi$
$733$	−12107.2	−0.610082	−0.305041	+	0.952339i	$0.598670\pi$
$734$	0	0
$735$	−1853.34	−0.0930088
$736$	0	0
$737$	−55166.4	−2.75723
$738$	0	0
$739$	4506.27	0.224311	0.112156	−	0.993691i	$-0.464225\pi$
$739$	4506.27	0.224311	0.112156	+	0.993691i	$0.464225\pi$
$740$	0	0
$741$	−29782.9	−1.47652
$742$	0	0
$743$	−1177.19	−0.0581253	−0.0290626	−	0.999578i	$-0.509252\pi$
$743$	−1177.19	−0.0581253	−0.0290626	+	0.999578i	$0.509252\pi$
$744$	0	0
$745$	−1072.40	−0.0527378
$746$	0	0
$747$	−40694.6	−1.99322
$748$	0	0
$749$	−1111.46	−0.0542214
$750$	0	0
$751$	−27631.6	−1.34260	−0.671300	−	0.741186i	$-0.734264\pi$
$751$	−27631.6	−1.34260	−0.671300	+	0.741186i	$0.734264\pi$
$752$	0	0
$753$	−7418.59	−0.359028
$754$	0	0
$755$	79.8616	0.00384962
$756$	0	0
$757$	11336.6	0.544300	0.272150	−	0.962255i	$-0.412265\pi$
$757$	11336.6	0.544300	0.272150	+	0.962255i	$0.412265\pi$
$758$	0	0
$759$	−56684.6	−2.71083
$760$	0	0
$761$	4356.58	0.207524	0.103762	−	0.994602i	$-0.466912\pi$
$761$	4356.58	0.207524	0.103762	+	0.994602i	$0.466912\pi$
$762$	0	0
$763$	−6376.37	−0.302543
$764$	0	0
$765$	2128.26	0.100585
$766$	0	0
$767$	−4940.87	−0.232601
$768$	0	0
$769$	−21718.1	−1.01843	−0.509217	−	0.860638i	$-0.670065\pi$
$769$	−21718.1	−1.01843	−0.509217	+	0.860638i	$0.670065\pi$
$770$	0	0
$771$	−41260.0	−1.92729
$772$	0	0
$773$	22688.4	1.05568	0.527842	−	0.849343i	$-0.323001\pi$
$773$	22688.4	1.05568	0.527842	+	0.849343i	$0.323001\pi$
$774$	0	0
$775$	−29410.1	−1.36315
$776$	0	0
$777$	−4360.13	−0.201311
$778$	0	0
$779$	13930.7	0.640716
$780$	0	0
$781$	24698.6	1.13161
$782$	0	0
$783$	8423.43	0.384456
$784$	0	0
$785$	−494.718	−0.0224933
$786$	0	0
$787$	32890.9	1.48975	0.744875	−	0.667204i	$-0.232509\pi$
$787$	32890.9	1.48975	0.744875	+	0.667204i	$0.232509\pi$
$788$	0	0
$789$	35262.6	1.59111
$790$	0	0
$791$	1134.85	0.0510123
$792$	0	0
$793$	36119.3	1.61744
$794$	0	0
$795$	1994.14	0.0889620
$796$	0	0
$797$	30404.1	1.35128	0.675638	−	0.737233i	$-0.263868\pi$
$797$	30404.1	1.35128	0.675638	+	0.737233i	$0.263868\pi$
$798$	0	0
$799$	19838.9	0.878410
$800$	0	0
$801$	65333.5	2.88196
$802$	0	0
$803$	−37960.7	−1.66825
$804$	0	0
$805$	378.767	0.0165836
$806$	0	0
$807$	41813.5	1.82392
$808$	0	0
$809$	−37889.3	−1.64662	−0.823311	−	0.567591i	$-0.807876\pi$
$809$	−37889.3	−1.64662	−0.823311	+	0.567591i	$0.807876\pi$
$810$	0	0
$811$	8123.23	0.351720	0.175860	−	0.984415i	$-0.443729\pi$
$811$	8123.23	0.351720	0.175860	+	0.984415i	$0.443729\pi$
$812$	0	0
$813$	−36625.3	−1.57996
$814$	0	0
$815$	−353.333	−0.0151862
$816$	0	0
$817$	−5233.39	−0.224104
$818$	0	0
$819$	17860.4	0.762020
$820$	0	0
$821$	13226.8	0.562264	0.281132	−	0.959669i	$-0.409290\pi$
$821$	13226.8	0.562264	0.281132	+	0.959669i	$0.409290\pi$
$822$	0	0
$823$	29575.2	1.25265	0.626323	−	0.779563i	$-0.284559\pi$
$823$	29575.2	1.25265	0.626323	+	0.779563i	$0.284559\pi$
$824$	0	0
$825$	−75212.5	−3.17401
$826$	0	0
$827$	36661.2	1.54152	0.770758	−	0.637128i	$-0.219878\pi$
$827$	36661.2	1.54152	0.770758	+	0.637128i	$0.219878\pi$
$828$	0	0
$829$	11277.0	0.472455	0.236228	−	0.971698i	$-0.424089\pi$
$829$	11277.0	0.472455	0.236228	+	0.971698i	$0.424089\pi$
$830$	0	0
$831$	20496.4	0.855611
$832$	0	0
$833$	−16929.2	−0.704158
$834$	0	0
$835$	−318.320	−0.0131927
$836$	0	0
$837$	68577.0	2.83198
$838$	0	0
$839$	18965.7	0.780417	0.390208	−	0.920727i	$-0.372403\pi$
$839$	18965.7	0.780417	0.390208	+	0.920727i	$0.372403\pi$
$840$	0	0
$841$	841.000	0.0344828
$842$	0	0
$843$	−23688.5	−0.967825
$844$	0	0
$845$	183.925	0.00748781
$846$	0	0
$847$	−18036.2	−0.731679
$848$	0	0
$849$	35762.5	1.44566
$850$	0	0
$851$	−7210.54	−0.290451
$852$	0	0
$853$	8067.23	0.323818	0.161909	−	0.986806i	$-0.448235\pi$
$853$	8067.23	0.323818	0.161909	+	0.986806i	$0.448235\pi$
$854$	0	0
$855$	2484.77	0.0993886
$856$	0	0
$857$	15281.7	0.609118	0.304559	−	0.952493i	$-0.401491\pi$
$857$	15281.7	0.609118	0.304559	+	0.952493i	$0.401491\pi$
$858$	0	0
$859$	36789.1	1.46127	0.730634	−	0.682770i	$-0.239225\pi$
$859$	36789.1	1.46127	0.730634	+	0.682770i	$0.239225\pi$
$860$	0	0
$861$	−12214.6	−0.483476
$862$	0	0
$863$	−40907.8	−1.61358	−0.806788	−	0.590840i	$-0.798796\pi$
$863$	−40907.8	−1.61358	−0.806788	+	0.590840i	$0.798796\pi$
$864$	0	0
$865$	−2147.71	−0.0844213
$866$	0	0
$867$	−16984.7	−0.665319
$868$	0	0
$869$	23076.2	0.900812
$870$	0	0
$871$	42029.4	1.63503
$872$	0	0
$873$	−47057.5	−1.82435
$874$	0	0
$875$	1006.88	0.0389015
$876$	0	0
$877$	2391.60	0.0920852	0.0460426	−	0.998939i	$-0.485339\pi$
$877$	2391.60	0.0920852	0.0460426	+	0.998939i	$0.485339\pi$
$878$	0	0
$879$	−35896.8	−1.37744
$880$	0	0
$881$	5487.72	0.209859	0.104930	−	0.994480i	$-0.466538\pi$
$881$	5487.72	0.209859	0.104930	+	0.994480i	$0.466538\pi$
$882$	0	0
$883$	−170.008	−0.00647931	−0.00323966	−	0.999995i	$-0.501031\pi$
$883$	−170.008	−0.00647931	−0.00323966	+	0.999995i	$0.501031\pi$
$884$	0	0
$885$	602.705	0.0228923
$886$	0	0
$887$	−25867.3	−0.979188	−0.489594	−	0.871950i	$-0.662855\pi$
$887$	−25867.3	−0.979188	−0.489594	+	0.871950i	$0.662855\pi$
$888$	0	0
$889$	12281.7	0.463345
$890$	0	0
$891$	72324.4	2.71937
$892$	0	0
$893$	23162.1	0.867961
$894$	0	0
$895$	369.327	0.0137936
$896$	0	0
$897$	43186.0	1.60751
$898$	0	0
$899$	6846.77	0.254007
$900$	0	0
$901$	18215.4	0.673520
$902$	0	0
$903$	4588.71	0.169106
$904$	0	0
$905$	−990.093	−0.0363666
$906$	0	0
$907$	−11411.5	−0.417765	−0.208882	−	0.977941i	$-0.566983\pi$
$907$	−11411.5	−0.417765	−0.208882	+	0.977941i	$0.566983\pi$
$908$	0	0
$909$	−80277.3	−2.92919
$910$	0	0
$911$	38718.5	1.40813	0.704063	−	0.710138i	$-0.251367\pi$
$911$	38718.5	1.40813	0.704063	+	0.710138i	$0.251367\pi$
$912$	0	0
$913$	−45500.1	−1.64933
$914$	0	0
$915$	−4405.96	−0.159187
$916$	0	0
$917$	3448.62	0.124191
$918$	0	0
$919$	48465.3	1.73963	0.869817	−	0.493374i	$-0.164237\pi$
$919$	48465.3	1.73963	0.869817	+	0.493374i	$0.164237\pi$
$920$	0	0
$921$	3726.72	0.133333
$922$	0	0
$923$	−18817.0	−0.671040
$924$	0	0
$925$	−9567.37	−0.340079
$926$	0	0
$927$	37081.6	1.31383
$928$	0	0
$929$	10560.1	0.372946	0.186473	−	0.982460i	$-0.440294\pi$
$929$	10560.1	0.372946	0.186473	+	0.982460i	$0.440294\pi$
$930$	0	0
$931$	−19765.0	−0.695782
$932$	0	0
$933$	44722.4	1.56929
$934$	0	0
$935$	2379.59	0.0832309
$936$	0	0
$937$	−23025.0	−0.802769	−0.401384	−	0.915910i	$-0.631471\pi$
$937$	−23025.0	−0.802769	−0.401384	+	0.915910i	$0.631471\pi$
$938$	0	0
$939$	−78971.7	−2.74456
$940$	0	0
$941$	40778.1	1.41268	0.706338	−	0.707874i	$-0.250346\pi$
$941$	40778.1	1.41268	0.706338	+	0.707874i	$0.250346\pi$
$942$	0	0
$943$	−20199.8	−0.697558
$944$	0	0
$945$	−1171.87	−0.0403396
$946$	0	0
$947$	−36129.8	−1.23977	−0.619884	−	0.784693i	$-0.712821\pi$
$947$	−36129.8	−1.23977	−0.619884	+	0.784693i	$0.712821\pi$
$948$	0	0
$949$	28920.9	0.989265
$950$	0	0
$951$	−16387.8	−0.558790
$952$	0	0
$953$	20831.4	0.708075	0.354037	−	0.935231i	$-0.384809\pi$
$953$	20831.4	0.708075	0.354037	+	0.935231i	$0.384809\pi$
$954$	0	0
$955$	−2976.86	−0.100868
$956$	0	0
$957$	17509.7	0.591441
$958$	0	0
$959$	1538.54	0.0518061
$960$	0	0
$961$	25950.1	0.871071
$962$	0	0
$963$	10572.6	0.353788
$964$	0	0
$965$	−1927.98	−0.0643149
$966$	0	0
$967$	−49242.5	−1.63757	−0.818785	−	0.574100i	$-0.805352\pi$
$967$	−49242.5	−1.63757	−0.818785	+	0.574100i	$0.805352\pi$
$968$	0	0
$969$	33185.7	1.10018
$970$	0	0
$971$	2352.05	0.0777351	0.0388675	−	0.999244i	$-0.487625\pi$
$971$	2352.05	0.0777351	0.0388675	+	0.999244i	$0.487625\pi$
$972$	0	0
$973$	−1487.89	−0.0490233
$974$	0	0
$975$	57301.8	1.88218
$976$	0	0
$977$	18768.3	0.614588	0.307294	−	0.951615i	$-0.400577\pi$
$977$	18768.3	0.614588	0.307294	+	0.951615i	$0.400577\pi$
$978$	0	0
$979$	73048.7	2.38473
$980$	0	0
$981$	60654.5	1.97406
$982$	0	0
$983$	49014.5	1.59036	0.795179	−	0.606375i	$-0.207377\pi$
$983$	49014.5	1.59036	0.795179	+	0.606375i	$0.207377\pi$
$984$	0	0
$985$	1761.81	0.0569908
$986$	0	0
$987$	−20308.9	−0.654953
$988$	0	0
$989$	7588.55	0.243986
$990$	0	0
$991$	48860.6	1.56620	0.783102	−	0.621893i	$-0.213636\pi$
$991$	48860.6	1.56620	0.783102	+	0.621893i	$0.213636\pi$
$992$	0	0
$993$	−7411.44	−0.236853
$994$	0	0
$995$	425.888	0.0135694
$996$	0	0
$997$	2934.57	0.0932184	0.0466092	−	0.998913i	$-0.485158\pi$
$997$	2934.57	0.0932184	0.0466092	+	0.998913i	$0.485158\pi$
$998$	0	0
$999$	22308.7	0.706524

Twists

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

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

\(f(q)\)	\(=\)	\(q+9.24264 q^{3} +0.656854 q^{5} -6.14214 q^{7} +58.4264 q^{9} +O(q^{10})\)	\(q+9.24264 q^{3} +0.656854 q^{5} -6.14214 q^{7} +58.4264 q^{9} +65.3259 q^{11} -49.7696 q^{13} +6.07107 q^{15} +55.4558 q^{17} +64.7452 q^{19} -56.7696 q^{21} -93.8823 q^{23} -124.569 q^{25} +290.463 q^{27} +29.0000 q^{29} +236.095 q^{31} +603.784 q^{33} -4.03449 q^{35} +76.8040 q^{37} -460.002 q^{39} +215.161 q^{41} -80.8305 q^{43} +38.3776 q^{45} +357.742 q^{47} -305.274 q^{49} +512.558 q^{51} +328.466 q^{53} +42.9096 q^{55} +598.416 q^{57} +99.2750 q^{59} -725.730 q^{61} -358.863 q^{63} -32.6913 q^{65} -844.479 q^{67} -867.720 q^{69} +378.083 q^{71} -581.097 q^{73} -1151.34 q^{75} -401.241 q^{77} +353.247 q^{79} +1107.13 q^{81} -696.510 q^{83} +36.4264 q^{85} +268.037 q^{87} +1118.22 q^{89} +305.691 q^{91} +2182.15 q^{93} +42.5281 q^{95} -805.415 q^{97} +3816.76 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 10 q^{3} - 10 q^{5} + 16 q^{7} + 32 q^{9}+O(q^{10}) \) 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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