LMFDB - Embedded newform 1344.4.a.bu.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1344,4,Mod(1,1344)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1344, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("1344.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1344 = 2^{6} \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1344.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:	$79.2985670477$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.37341.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 57x - 148 $ x^3 - 57*x - 148
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 672)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$8.61298$ of defining polynomial
Character	$\chi$	$=$	1344.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.00000 q^{3} +15.2260 q^{5} -7.00000 q^{7} +9.00000 q^{9} +O(q^{10})$	$q+3.00000 q^{3} +15.2260 q^{5} -7.00000 q^{7} +9.00000 q^{9} +26.3001 q^{11} +73.8297 q^{13} +45.6779 q^{15} +7.69991 q^{17} +69.3778 q^{19} -21.0000 q^{21} -74.6037 q^{23} +106.830 q^{25} +27.0000 q^{27} +145.482 q^{29} -79.2295 q^{31} +78.9003 q^{33} -106.582 q^{35} +5.82968 q^{37} +221.489 q^{39} +203.352 q^{41} -95.6594 q^{43} +137.034 q^{45} -471.875 q^{47} +49.0000 q^{49} +23.0997 q^{51} -361.489 q^{53} +400.444 q^{55} +208.133 q^{57} +834.586 q^{59} -734.298 q^{61} -63.0000 q^{63} +1124.13 q^{65} -624.735 q^{67} -223.811 q^{69} -202.819 q^{71} +830.881 q^{73} +320.489 q^{75} -184.101 q^{77} +848.468 q^{79} +81.0000 q^{81} +778.297 q^{83} +117.238 q^{85} +436.446 q^{87} +400.773 q^{89} -516.808 q^{91} -237.688 q^{93} +1056.34 q^{95} +119.015 q^{97} +236.701 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 9 q^{3} - 6 q^{5} - 21 q^{7} + 27 q^{9}+O(q^{10}) $ 3 * q + 9 * q^3 - 6 * q^5 - 21 * q^7 + 27 * q^9	$ 3 q + 9 q^{3} - 6 q^{5} - 21 q^{7} + 27 q^{9} + 48 q^{11} - 6 q^{13} - 18 q^{15} + 54 q^{17} + 84 q^{19} - 63 q^{21} - 48 q^{23} + 93 q^{25} + 81 q^{27} - 18 q^{29} - 72 q^{31} + 144 q^{33} + 42 q^{35} - 210 q^{37} - 18 q^{39} + 414 q^{41} + 168 q^{43} - 54 q^{45} - 72 q^{47} + 147 q^{49} + 162 q^{51} - 402 q^{53} + 456 q^{55} + 252 q^{57} + 540 q^{59} - 798 q^{61} - 189 q^{63} + 1740 q^{65} + 48 q^{67} - 144 q^{69} + 456 q^{71} + 1230 q^{73} + 279 q^{75} - 336 q^{77} + 1368 q^{79} + 243 q^{81} + 60 q^{83} - 660 q^{85} - 54 q^{87} + 2742 q^{89} + 42 q^{91} - 216 q^{93} + 648 q^{95} + 1950 q^{97} + 432 q^{99}+O(q^{100}) $ 3 * q + 9 * q^3 - 6 * q^5 - 21 * q^7 + 27 * q^9 + 48 * q^11 - 6 * q^13 - 18 * q^15 + 54 * q^17 + 84 * q^19 - 63 * q^21 - 48 * q^23 + 93 * q^25 + 81 * q^27 - 18 * q^29 - 72 * q^31 + 144 * q^33 + 42 * q^35 - 210 * q^37 - 18 * q^39 + 414 * q^41 + 168 * q^43 - 54 * q^45 - 72 * q^47 + 147 * q^49 + 162 * q^51 - 402 * q^53 + 456 * q^55 + 252 * q^57 + 540 * q^59 - 798 * q^61 - 189 * q^63 + 1740 * q^65 + 48 * q^67 - 144 * q^69 + 456 * q^71 + 1230 * q^73 + 279 * q^75 - 336 * q^77 + 1368 * q^79 + 243 * q^81 + 60 * q^83 - 660 * q^85 - 54 * q^87 + 2742 * q^89 + 42 * q^91 - 216 * q^93 + 648 * q^95 + 1950 * q^97 + 432 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	0	0
$5$	15.2260	1.36185	0.680925	−	0.732353i	$-0.261578\pi$
$5$	15.2260	1.36185	0.680925	+	0.732353i	$0.261578\pi$
$6$	0	0
$7$	−7.00000	−0.377964
$7$	−7.00000	−0.377964
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	26.3001	0.720889	0.360444	−	0.932781i	$-0.382625\pi$
$11$	26.3001	0.720889	0.360444	+	0.932781i	$0.382625\pi$
$12$	0	0
$13$	73.8297	1.57513	0.787564	−	0.616233i	$-0.211342\pi$
$13$	73.8297	1.57513	0.787564	+	0.616233i	$0.211342\pi$
$14$	0	0
$15$	45.6779	0.786265
$16$	0	0
$17$	7.69991	0.109853	0.0549265	−	0.998490i	$-0.482508\pi$
$17$	7.69991	0.109853	0.0549265	+	0.998490i	$0.482508\pi$
$18$	0	0
$19$	69.3778	0.837703	0.418851	−	0.908055i	$-0.362433\pi$
$19$	69.3778	0.837703	0.418851	+	0.908055i	$0.362433\pi$
$20$	0	0
$21$	−21.0000	−0.218218
$22$	0	0
$23$	−74.6037	−0.676346	−0.338173	−	0.941084i	$-0.609809\pi$
$23$	−74.6037	−0.676346	−0.338173	+	0.941084i	$0.609809\pi$
$24$	0	0
$25$	106.830	0.854637
$26$	0	0
$27$	27.0000	0.192450
$28$	0	0
$29$	145.482	0.931563	0.465781	−	0.884900i	$-0.345773\pi$
$29$	145.482	0.931563	0.465781	+	0.884900i	$0.345773\pi$
$30$	0	0
$31$	−79.2295	−0.459033	−0.229517	−	0.973305i	$-0.573715\pi$
$31$	−79.2295	−0.459033	−0.229517	+	0.973305i	$0.573715\pi$
$32$	0	0
$33$	78.9003	0.416205
$34$	0	0
$35$	−106.582	−0.514731
$36$	0	0
$37$	5.82968	0.0259025	0.0129513	−	0.999916i	$-0.495877\pi$
$37$	5.82968	0.0259025	0.0129513	+	0.999916i	$0.495877\pi$
$38$	0	0
$39$	221.489	0.909401
$40$	0	0
$41$	203.352	0.774592	0.387296	−	0.921955i	$-0.373409\pi$
$41$	203.352	0.774592	0.387296	+	0.921955i	$0.373409\pi$
$42$	0	0
$43$	−95.6594	−0.339254	−0.169627	−	0.985508i	$-0.554256\pi$
$43$	−95.6594	−0.339254	−0.169627	+	0.985508i	$0.554256\pi$
$44$	0	0
$45$	137.034	0.453950
$46$	0	0
$47$	−471.875	−1.46447	−0.732234	−	0.681053i	$-0.761522\pi$
$47$	−471.875	−1.46447	−0.732234	+	0.681053i	$0.761522\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	0	0
$51$	23.0997	0.0634237
$52$	0	0
$53$	−361.489	−0.936874	−0.468437	−	0.883497i	$-0.655183\pi$
$53$	−361.489	−0.936874	−0.468437	+	0.883497i	$0.655183\pi$
$54$	0	0
$55$	400.444	0.981743
$56$	0	0
$57$	208.133	0.483648
$58$	0	0
$59$	834.586	1.84159	0.920796	−	0.390046i	$-0.127541\pi$
$59$	834.586	1.84159	0.920796	+	0.390046i	$0.127541\pi$
$60$	0	0
$61$	−734.298	−1.54127	−0.770633	−	0.637280i	$-0.780060\pi$
$61$	−734.298	−1.54127	−0.770633	+	0.637280i	$0.780060\pi$
$62$	0	0
$63$	−63.0000	−0.125988
$64$	0	0
$65$	1124.13	2.14509
$66$	0	0
$67$	−624.735	−1.13916	−0.569578	−	0.821937i	$-0.692893\pi$
$67$	−624.735	−1.13916	−0.569578	+	0.821937i	$0.692893\pi$
$68$	0	0
$69$	−223.811	−0.390488
$70$	0	0
$71$	−202.819	−0.339017	−0.169509	−	0.985529i	$-0.554218\pi$
$71$	−202.819	−0.339017	−0.169509	+	0.985529i	$0.554218\pi$
$72$	0	0
$73$	830.881	1.33215	0.666077	−	0.745883i	$-0.267972\pi$
$73$	830.881	1.33215	0.666077	+	0.745883i	$0.267972\pi$
$74$	0	0
$75$	320.489	0.493425
$76$	0	0
$77$	−184.101	−0.272470
$78$	0	0
$79$	848.468	1.20836	0.604178	−	0.796850i	$-0.293502\pi$
$79$	848.468	1.20836	0.604178	+	0.796850i	$0.293502\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	778.297	1.02927	0.514634	−	0.857410i	$-0.327928\pi$
$83$	778.297	1.02927	0.514634	+	0.857410i	$0.327928\pi$
$84$	0	0
$85$	117.238	0.149603
$86$	0	0
$87$	436.446	0.537838
$88$	0	0
$89$	400.773	0.477324	0.238662	−	0.971103i	$-0.423291\pi$
$89$	400.773	0.477324	0.238662	+	0.971103i	$0.423291\pi$
$90$	0	0
$91$	−516.808	−0.595343
$92$	0	0
$93$	−237.688	−0.265023
$94$	0	0
$95$	1056.34	1.14083
$96$	0	0
$97$	119.015	0.124579	0.0622893	−	0.998058i	$-0.480160\pi$
$97$	119.015	0.124579	0.0622893	+	0.998058i	$0.480160\pi$
$98$	0	0
$99$	236.701	0.240296
$100$	0	0
$101$	797.233	0.785423	0.392711	−	0.919662i	$-0.371537\pi$
$101$	797.233	0.785423	0.392711	+	0.919662i	$0.371537\pi$
$102$	0	0
$103$	−317.943	−0.304154	−0.152077	−	0.988369i	$-0.548596\pi$
$103$	−317.943	−0.304154	−0.152077	+	0.988369i	$0.548596\pi$
$104$	0	0
$105$	−319.745	−0.297180
$106$	0	0
$107$	−1001.26	−0.904626	−0.452313	−	0.891859i	$-0.649401\pi$
$107$	−1001.26	−0.904626	−0.452313	+	0.891859i	$0.649401\pi$
$108$	0	0
$109$	281.319	0.247206	0.123603	−	0.992332i	$-0.460555\pi$
$109$	281.319	0.247206	0.123603	+	0.992332i	$0.460555\pi$
$110$	0	0
$111$	17.4890	0.0149548
$112$	0	0
$113$	1330.35	1.10752	0.553758	−	0.832678i	$-0.313193\pi$
$113$	1330.35	1.10752	0.553758	+	0.832678i	$0.313193\pi$
$114$	0	0
$115$	−1135.91	−0.921082
$116$	0	0
$117$	664.467	0.525043
$118$	0	0
$119$	−53.8993	−0.0415205
$120$	0	0
$121$	−639.305	−0.480319
$122$	0	0
$123$	610.057	0.447211
$124$	0	0
$125$	−276.661	−0.197962
$126$	0	0
$127$	−2391.57	−1.67100	−0.835501	−	0.549489i	$-0.814822\pi$
$127$	−2391.57	−1.67100	−0.835501	+	0.549489i	$0.814822\pi$
$128$	0	0
$129$	−286.978	−0.195868
$130$	0	0
$131$	1924.00	1.28321	0.641607	−	0.767034i	$-0.278268\pi$
$131$	1924.00	1.28321	0.641607	+	0.767034i	$0.278268\pi$
$132$	0	0
$133$	−485.644	−0.316622
$134$	0	0
$135$	411.101	0.262088
$136$	0	0
$137$	−1213.71	−0.756895	−0.378448	−	0.925623i	$-0.623542\pi$
$137$	−1213.71	−0.756895	−0.378448	+	0.925623i	$0.623542\pi$
$138$	0	0
$139$	−179.112	−0.109296	−0.0546478	−	0.998506i	$-0.517404\pi$
$139$	−179.112	−0.109296	−0.0546478	+	0.998506i	$0.517404\pi$
$140$	0	0
$141$	−1415.62	−0.845511
$142$	0	0
$143$	1941.73	1.13549
$144$	0	0
$145$	2215.10	1.26865
$146$	0	0
$147$	147.000	0.0824786
$148$	0	0
$149$	−628.035	−0.345306	−0.172653	−	0.984983i	$-0.555234\pi$
$149$	−628.035	−0.345306	−0.172653	+	0.984983i	$0.555234\pi$
$150$	0	0
$151$	916.828	0.494108	0.247054	−	0.969002i	$-0.420537\pi$
$151$	916.828	0.494108	0.247054	+	0.969002i	$0.420537\pi$
$152$	0	0
$153$	69.2992	0.0366177
$154$	0	0
$155$	−1206.34	−0.625135
$156$	0	0
$157$	−2612.18	−1.32786	−0.663932	−	0.747793i	$-0.731114\pi$
$157$	−2612.18	−1.32786	−0.663932	+	0.747793i	$0.731114\pi$
$158$	0	0
$159$	−1084.47	−0.540905
$160$	0	0
$161$	522.226	0.255635
$162$	0	0
$163$	−2604.81	−1.25168	−0.625842	−	0.779950i	$-0.715244\pi$
$163$	−2604.81	−1.25168	−0.625842	+	0.779950i	$0.715244\pi$
$164$	0	0
$165$	1201.33	0.566810
$166$	0	0
$167$	−2222.20	−1.02969	−0.514847	−	0.857282i	$-0.672151\pi$
$167$	−2222.20	−1.02969	−0.514847	+	0.857282i	$0.672151\pi$
$168$	0	0
$169$	3253.82	1.48103
$170$	0	0
$171$	624.400	0.279234
$172$	0	0
$173$	−852.621	−0.374703	−0.187351	−	0.982293i	$-0.559990\pi$
$173$	−852.621	−0.374703	−0.187351	+	0.982293i	$0.559990\pi$
$174$	0	0
$175$	−747.808	−0.323023
$176$	0	0
$177$	2503.76	1.06324
$178$	0	0
$179$	−2103.08	−0.878164	−0.439082	−	0.898447i	$-0.644696\pi$
$179$	−2103.08	−0.878164	−0.439082	+	0.898447i	$0.644696\pi$
$180$	0	0
$181$	4634.86	1.90335	0.951674	−	0.307109i	$-0.0993616\pi$
$181$	4634.86	1.90335	0.951674	+	0.307109i	$0.0993616\pi$
$182$	0	0
$183$	−2202.89	−0.889850
$184$	0	0
$185$	88.7624	0.0352754
$186$	0	0
$187$	202.508	0.0791918
$188$	0	0
$189$	−189.000	−0.0727393
$190$	0	0
$191$	−600.460	−0.227475	−0.113738	−	0.993511i	$-0.536282\pi$
$191$	−600.460	−0.227475	−0.113738	+	0.993511i	$0.536282\pi$
$192$	0	0
$193$	3160.55	1.17876	0.589381	−	0.807855i	$-0.299372\pi$
$193$	3160.55	1.17876	0.589381	+	0.807855i	$0.299372\pi$
$194$	0	0
$195$	3372.38	1.23847
$196$	0	0
$197$	3986.23	1.44166	0.720830	−	0.693112i	$-0.243761\pi$
$197$	3986.23	1.44166	0.720830	+	0.693112i	$0.243761\pi$
$198$	0	0
$199$	4168.17	1.48479	0.742396	−	0.669961i	$-0.233689\pi$
$199$	4168.17	1.48479	0.742396	+	0.669961i	$0.233689\pi$
$200$	0	0
$201$	−1874.20	−0.657692
$202$	0	0
$203$	−1018.37	−0.352098
$204$	0	0
$205$	3096.23	1.05488
$206$	0	0
$207$	−671.434	−0.225449
$208$	0	0
$209$	1824.64	0.603890
$210$	0	0
$211$	1314.73	0.428956	0.214478	−	0.976729i	$-0.431195\pi$
$211$	1314.73	0.428956	0.214478	+	0.976729i	$0.431195\pi$
$212$	0	0
$213$	−608.458	−0.195732
$214$	0	0
$215$	−1456.50	−0.462013
$216$	0	0
$217$	554.606	0.173498
$218$	0	0
$219$	2492.64	0.769120
$220$	0	0
$221$	568.482	0.173033
$222$	0	0
$223$	−4532.44	−1.36105	−0.680526	−	0.732724i	$-0.738249\pi$
$223$	−4532.44	−1.36105	−0.680526	+	0.732724i	$0.738249\pi$
$224$	0	0
$225$	961.467	0.284879
$226$	0	0
$227$	4220.24	1.23395	0.616976	−	0.786982i	$-0.288358\pi$
$227$	4220.24	1.23395	0.616976	+	0.786982i	$0.288358\pi$
$228$	0	0
$229$	6129.58	1.76880	0.884398	−	0.466734i	$-0.154569\pi$
$229$	6129.58	1.76880	0.884398	+	0.466734i	$0.154569\pi$
$230$	0	0
$231$	−552.302	−0.157311
$232$	0	0
$233$	3700.68	1.04051	0.520256	−	0.854010i	$-0.325836\pi$
$233$	3700.68	1.04051	0.520256	+	0.854010i	$0.325836\pi$
$234$	0	0
$235$	−7184.74	−1.99439
$236$	0	0
$237$	2545.40	0.697644
$238$	0	0
$239$	−6293.01	−1.70318	−0.851591	−	0.524206i	$-0.824362\pi$
$239$	−6293.01	−1.70318	−0.851591	+	0.524206i	$0.824362\pi$
$240$	0	0
$241$	−1972.01	−0.527087	−0.263544	−	0.964647i	$-0.584891\pi$
$241$	−1972.01	−0.527087	−0.263544	+	0.964647i	$0.584891\pi$
$242$	0	0
$243$	243.000	0.0641500
$244$	0	0
$245$	746.072	0.194550
$246$	0	0
$247$	5122.14	1.31949
$248$	0	0
$249$	2334.89	0.594248
$250$	0	0
$251$	−5329.34	−1.34018	−0.670090	−	0.742280i	$-0.733744\pi$
$251$	−5329.34	−1.34018	−0.670090	+	0.742280i	$0.733744\pi$
$252$	0	0
$253$	−1962.08	−0.487570
$254$	0	0
$255$	351.715	0.0863736
$256$	0	0
$257$	3342.24	0.811218	0.405609	−	0.914047i	$-0.367060\pi$
$257$	3342.24	0.811218	0.405609	+	0.914047i	$0.367060\pi$
$258$	0	0
$259$	−40.8077	−0.00979023
$260$	0	0
$261$	1309.34	0.310521
$262$	0	0
$263$	995.408	0.233382	0.116691	−	0.993168i	$-0.462771\pi$
$263$	995.408	0.233382	0.116691	+	0.993168i	$0.462771\pi$
$264$	0	0
$265$	−5504.02	−1.27588
$266$	0	0
$267$	1202.32	0.275583
$268$	0	0
$269$	6289.66	1.42560	0.712802	−	0.701365i	$-0.247426\pi$
$269$	6289.66	1.42560	0.712802	+	0.701365i	$0.247426\pi$
$270$	0	0
$271$	7465.36	1.67339	0.836694	−	0.547671i	$-0.184485\pi$
$271$	7465.36	1.67339	0.836694	+	0.547671i	$0.184485\pi$
$272$	0	0
$273$	−1550.42	−0.343721
$274$	0	0
$275$	2809.63	0.616099
$276$	0	0
$277$	4275.95	0.927498	0.463749	−	0.885967i	$-0.346504\pi$
$277$	4275.95	0.927498	0.463749	+	0.885967i	$0.346504\pi$
$278$	0	0
$279$	−713.065	−0.153011
$280$	0	0
$281$	−2881.34	−0.611694	−0.305847	−	0.952081i	$-0.598940\pi$
$281$	−2881.34	−0.611694	−0.305847	+	0.952081i	$0.598940\pi$
$282$	0	0
$283$	6813.95	1.43126	0.715631	−	0.698478i	$-0.246139\pi$
$283$	6813.95	1.43126	0.715631	+	0.698478i	$0.246139\pi$
$284$	0	0
$285$	3169.03	0.658656
$286$	0	0
$287$	−1423.47	−0.292768
$288$	0	0
$289$	−4853.71	−0.987932
$290$	0	0
$291$	357.045	0.0719255
$292$	0	0
$293$	8241.56	1.64327	0.821633	−	0.570016i	$-0.193063\pi$
$293$	8241.56	1.64327	0.821633	+	0.570016i	$0.193063\pi$
$294$	0	0
$295$	12707.4	2.50797
$296$	0	0
$297$	710.103	0.138735
$298$	0	0
$299$	−5507.97	−1.06533
$300$	0	0
$301$	669.615	0.128226
$302$	0	0
$303$	2391.70	0.453464
$304$	0	0
$305$	−11180.4	−2.09897
$306$	0	0
$307$	−6818.59	−1.26761	−0.633807	−	0.773491i	$-0.718509\pi$
$307$	−6818.59	−1.26761	−0.633807	+	0.773491i	$0.718509\pi$
$308$	0	0
$309$	−953.829	−0.175603
$310$	0	0
$311$	−8875.34	−1.61825	−0.809123	−	0.587639i	$-0.800057\pi$
$311$	−8875.34	−1.61825	−0.809123	+	0.587639i	$0.800057\pi$
$312$	0	0
$313$	5587.37	1.00900	0.504499	−	0.863412i	$-0.331677\pi$
$313$	5587.37	1.00900	0.504499	+	0.863412i	$0.331677\pi$
$314$	0	0
$315$	−959.235	−0.171577
$316$	0	0
$317$	−5185.09	−0.918686	−0.459343	−	0.888259i	$-0.651915\pi$
$317$	−5185.09	−0.918686	−0.459343	+	0.888259i	$0.651915\pi$
$318$	0	0
$319$	3826.19	0.671553
$320$	0	0
$321$	−3003.77	−0.522286
$322$	0	0
$323$	534.202	0.0920242
$324$	0	0
$325$	7887.20	1.34616
$326$	0	0
$327$	843.956	0.142724
$328$	0	0
$329$	3303.12	0.553517
$330$	0	0
$331$	9657.71	1.60373	0.801866	−	0.597503i	$-0.203841\pi$
$331$	9657.71	1.60373	0.801866	+	0.597503i	$0.203841\pi$
$332$	0	0
$333$	52.4671	0.00863417
$334$	0	0
$335$	−9512.18	−1.55136
$336$	0	0
$337$	−3392.37	−0.548350	−0.274175	−	0.961680i	$-0.588405\pi$
$337$	−3392.37	−0.548350	−0.274175	+	0.961680i	$0.588405\pi$
$338$	0	0
$339$	3991.06	0.639424
$340$	0	0
$341$	−2083.74	−0.330912
$342$	0	0
$343$	−343.000	−0.0539949
$344$	0	0
$345$	−3407.74	−0.531787
$346$	0	0
$347$	−8108.84	−1.25448	−0.627241	−	0.778825i	$-0.715816\pi$
$347$	−8108.84	−1.25448	−0.627241	+	0.778825i	$0.715816\pi$
$348$	0	0
$349$	−5485.46	−0.841346	−0.420673	−	0.907212i	$-0.638206\pi$
$349$	−5485.46	−0.841346	−0.420673	+	0.907212i	$0.638206\pi$
$350$	0	0
$351$	1993.40	0.303134
$352$	0	0
$353$	−7161.17	−1.07975	−0.539874	−	0.841746i	$-0.681528\pi$
$353$	−7161.17	−1.07975	−0.539874	+	0.841746i	$0.681528\pi$
$354$	0	0
$355$	−3088.12	−0.461691
$356$	0	0
$357$	−161.698	−0.0239719
$358$	0	0
$359$	−6181.38	−0.908749	−0.454374	−	0.890811i	$-0.650137\pi$
$359$	−6181.38	−0.908749	−0.454374	+	0.890811i	$0.650137\pi$
$360$	0	0
$361$	−2045.73	−0.298254
$362$	0	0
$363$	−1917.92	−0.277313
$364$	0	0
$365$	12651.0	1.81420
$366$	0	0
$367$	6525.78	0.928182	0.464091	−	0.885787i	$-0.346381\pi$
$367$	6525.78	0.928182	0.464091	+	0.885787i	$0.346381\pi$
$368$	0	0
$369$	1830.17	0.258197
$370$	0	0
$371$	2530.42	0.354105
$372$	0	0
$373$	−9218.61	−1.27968	−0.639841	−	0.768507i	$-0.721000\pi$
$373$	−9218.61	−1.27968	−0.639841	+	0.768507i	$0.721000\pi$
$374$	0	0
$375$	−829.982	−0.114293
$376$	0	0
$377$	10740.9	1.46733
$378$	0	0
$379$	−3621.61	−0.490843	−0.245421	−	0.969417i	$-0.578926\pi$
$379$	−3621.61	−0.490843	−0.245421	+	0.969417i	$0.578926\pi$
$380$	0	0
$381$	−7174.70	−0.964753
$382$	0	0
$383$	1673.51	0.223270	0.111635	−	0.993749i	$-0.464391\pi$
$383$	1673.51	0.223270	0.111635	+	0.993749i	$0.464391\pi$
$384$	0	0
$385$	−2803.11	−0.371064
$386$	0	0
$387$	−860.934	−0.113085
$388$	0	0
$389$	−9146.32	−1.19213	−0.596063	−	0.802938i	$-0.703269\pi$
$389$	−9146.32	−1.19213	−0.596063	+	0.802938i	$0.703269\pi$
$390$	0	0
$391$	−574.442	−0.0742986
$392$	0	0
$393$	5772.01	0.740864
$394$	0	0
$395$	12918.7	1.64560
$396$	0	0
$397$	3075.56	0.388811	0.194405	−	0.980921i	$-0.437722\pi$
$397$	3075.56	0.388811	0.194405	+	0.980921i	$0.437722\pi$
$398$	0	0
$399$	−1456.93	−0.182802
$400$	0	0
$401$	−6833.60	−0.851007	−0.425503	−	0.904957i	$-0.639903\pi$
$401$	−6833.60	−0.851007	−0.425503	+	0.904957i	$0.639903\pi$
$402$	0	0
$403$	−5849.49	−0.723037
$404$	0	0
$405$	1233.30	0.151317
$406$	0	0
$407$	153.321	0.0186728
$408$	0	0
$409$	6648.78	0.803817	0.401909	−	0.915680i	$-0.368347\pi$
$409$	6648.78	0.803817	0.401909	+	0.915680i	$0.368347\pi$
$410$	0	0
$411$	−3641.14	−0.436994
$412$	0	0
$413$	−5842.10	−0.696056
$414$	0	0
$415$	11850.3	1.40171
$416$	0	0
$417$	−537.336	−0.0631018
$418$	0	0
$419$	8068.93	0.940795	0.470397	−	0.882455i	$-0.344111\pi$
$419$	8068.93	0.940795	0.470397	+	0.882455i	$0.344111\pi$
$420$	0	0
$421$	−14024.0	−1.62349	−0.811744	−	0.584013i	$-0.801482\pi$
$421$	−14024.0	−1.62349	−0.811744	+	0.584013i	$0.801482\pi$
$422$	0	0
$423$	−4246.87	−0.488156
$424$	0	0
$425$	822.579	0.0938845
$426$	0	0
$427$	5140.08	0.582543
$428$	0	0
$429$	5825.18	0.655577
$430$	0	0
$431$	−9644.09	−1.07782	−0.538909	−	0.842364i	$-0.681163\pi$
$431$	−9644.09	−1.07782	−0.538909	+	0.842364i	$0.681163\pi$
$432$	0	0
$433$	−6827.98	−0.757810	−0.378905	−	0.925435i	$-0.623699\pi$
$433$	−6827.98	−0.757810	−0.378905	+	0.925435i	$0.623699\pi$
$434$	0	0
$435$	6645.30	0.732455
$436$	0	0
$437$	−5175.84	−0.566577
$438$	0	0
$439$	16801.2	1.82660	0.913299	−	0.407289i	$-0.133526\pi$
$439$	16801.2	1.82660	0.913299	+	0.407289i	$0.133526\pi$
$440$	0	0
$441$	441.000	0.0476190
$442$	0	0
$443$	−5950.24	−0.638159	−0.319079	−	0.947728i	$-0.603374\pi$
$443$	−5950.24	−0.638159	−0.319079	+	0.947728i	$0.603374\pi$
$444$	0	0
$445$	6102.15	0.650044
$446$	0	0
$447$	−1884.10	−0.199363
$448$	0	0
$449$	−12064.9	−1.26811	−0.634053	−	0.773289i	$-0.718610\pi$
$449$	−12064.9	−1.26811	−0.634053	+	0.773289i	$0.718610\pi$
$450$	0	0
$451$	5348.18	0.558395
$452$	0	0
$453$	2750.48	0.285274
$454$	0	0
$455$	−7868.89	−0.810768
$456$	0	0
$457$	−2786.02	−0.285174	−0.142587	−	0.989782i	$-0.545542\pi$
$457$	−2786.02	−0.285174	−0.142587	+	0.989782i	$0.545542\pi$
$458$	0	0
$459$	207.897	0.0211412
$460$	0	0
$461$	−7851.81	−0.793265	−0.396633	−	0.917977i	$-0.629821\pi$
$461$	−7851.81	−0.793265	−0.396633	+	0.917977i	$0.629821\pi$
$462$	0	0
$463$	−4455.88	−0.447262	−0.223631	−	0.974674i	$-0.571791\pi$
$463$	−4455.88	−0.447262	−0.223631	+	0.974674i	$0.571791\pi$
$464$	0	0
$465$	−3619.03	−0.360922
$466$	0	0
$467$	−11950.4	−1.18415	−0.592077	−	0.805881i	$-0.701692\pi$
$467$	−11950.4	−1.18415	−0.592077	+	0.805881i	$0.701692\pi$
$468$	0	0
$469$	4373.14	0.430561
$470$	0	0
$471$	−7836.55	−0.766643
$472$	0	0
$473$	−2515.85	−0.244564
$474$	0	0
$475$	7411.60	0.715932
$476$	0	0
$477$	−3253.40	−0.312291
$478$	0	0
$479$	6961.94	0.664090	0.332045	−	0.943263i	$-0.392261\pi$
$479$	6961.94	0.664090	0.332045	+	0.943263i	$0.392261\pi$
$480$	0	0
$481$	430.403	0.0407998
$482$	0	0
$483$	1566.68	0.147591
$484$	0	0
$485$	1812.11	0.169658
$486$	0	0
$487$	10606.0	0.986866	0.493433	−	0.869784i	$-0.335742\pi$
$487$	10606.0	0.986866	0.493433	+	0.869784i	$0.335742\pi$
$488$	0	0
$489$	−7814.43	−0.722660
$490$	0	0
$491$	1772.77	0.162941	0.0814704	−	0.996676i	$-0.474038\pi$
$491$	1772.77	0.162941	0.0814704	+	0.996676i	$0.474038\pi$
$492$	0	0
$493$	1120.20	0.102335
$494$	0	0
$495$	3604.00	0.327248
$496$	0	0
$497$	1419.73	0.128136
$498$	0	0
$499$	−4788.45	−0.429580	−0.214790	−	0.976660i	$-0.568907\pi$
$499$	−4788.45	−0.429580	−0.214790	+	0.976660i	$0.568907\pi$
$500$	0	0
$501$	−6666.59	−0.594494
$502$	0	0
$503$	−5635.24	−0.499529	−0.249764	−	0.968307i	$-0.580353\pi$
$503$	−5635.24	−0.499529	−0.249764	+	0.968307i	$0.580353\pi$
$504$	0	0
$505$	12138.6	1.06963
$506$	0	0
$507$	9761.46	0.855073
$508$	0	0
$509$	−9394.42	−0.818075	−0.409038	−	0.912518i	$-0.634135\pi$
$509$	−9394.42	−0.818075	−0.409038	+	0.912518i	$0.634135\pi$
$510$	0	0
$511$	−5816.17	−0.503507
$512$	0	0
$513$	1873.20	0.161216
$514$	0	0
$515$	−4840.99	−0.414212
$516$	0	0
$517$	−12410.4	−1.05572
$518$	0	0
$519$	−2557.86	−0.216335
$520$	0	0
$521$	6730.95	0.566004	0.283002	−	0.959119i	$-0.408670\pi$
$521$	6730.95	0.566004	0.283002	+	0.959119i	$0.408670\pi$
$522$	0	0
$523$	−16770.5	−1.40214	−0.701072	−	0.713090i	$-0.747295\pi$
$523$	−16770.5	−1.40214	−0.701072	+	0.713090i	$0.747295\pi$
$524$	0	0
$525$	−2243.42	−0.186497
$526$	0	0
$527$	−610.060	−0.0504262
$528$	0	0
$529$	−6601.28	−0.542556
$530$	0	0
$531$	7511.28	0.613864
$532$	0	0
$533$	15013.4	1.22008
$534$	0	0
$535$	−15245.1	−1.23197
$536$	0	0
$537$	−6309.23	−0.507008
$538$	0	0
$539$	1288.70	0.102984
$540$	0	0
$541$	−20179.2	−1.60364	−0.801822	−	0.597563i	$-0.796136\pi$
$541$	−20179.2	−1.60364	−0.801822	+	0.597563i	$0.796136\pi$
$542$	0	0
$543$	13904.6	1.09890
$544$	0	0
$545$	4283.35	0.336658
$546$	0	0
$547$	21151.5	1.65333	0.826667	−	0.562691i	$-0.190234\pi$
$547$	21151.5	1.65333	0.826667	+	0.562691i	$0.190234\pi$
$548$	0	0
$549$	−6608.68	−0.513755
$550$	0	0
$551$	10093.2	0.780373
$552$	0	0
$553$	−5939.28	−0.456716
$554$	0	0
$555$	266.287	0.0203662
$556$	0	0
$557$	−9359.89	−0.712013	−0.356007	−	0.934483i	$-0.615862\pi$
$557$	−9359.89	−0.712013	−0.356007	+	0.934483i	$0.615862\pi$
$558$	0	0
$559$	−7062.50	−0.534368
$560$	0	0
$561$	607.525	0.0457214
$562$	0	0
$563$	−17451.1	−1.30635	−0.653177	−	0.757205i	$-0.726564\pi$
$563$	−17451.1	−1.30635	−0.653177	+	0.757205i	$0.726564\pi$
$564$	0	0
$565$	20255.9	1.50827
$566$	0	0
$567$	−567.000	−0.0419961
$568$	0	0
$569$	9593.01	0.706783	0.353392	−	0.935475i	$-0.385028\pi$
$569$	9593.01	0.706783	0.353392	+	0.935475i	$0.385028\pi$
$570$	0	0
$571$	4038.62	0.295991	0.147995	−	0.988988i	$-0.452718\pi$
$571$	4038.62	0.295991	0.147995	+	0.988988i	$0.452718\pi$
$572$	0	0
$573$	−1801.38	−0.131333
$574$	0	0
$575$	−7969.89	−0.578030
$576$	0	0
$577$	3994.10	0.288174	0.144087	−	0.989565i	$-0.453975\pi$
$577$	3994.10	0.288174	0.144087	+	0.989565i	$0.453975\pi$
$578$	0	0
$579$	9481.64	0.680559
$580$	0	0
$581$	−5448.08	−0.389026
$582$	0	0
$583$	−9507.20	−0.675382
$584$	0	0
$585$	10117.1	0.715030
$586$	0	0
$587$	1661.76	0.116845	0.0584225	−	0.998292i	$-0.481393\pi$
$587$	1661.76	0.116845	0.0584225	+	0.998292i	$0.481393\pi$
$588$	0	0
$589$	−5496.77	−0.384534
$590$	0	0
$591$	11958.7	0.832343
$592$	0	0
$593$	85.1016	0.00589326	0.00294663	−	0.999996i	$-0.499062\pi$
$593$	85.1016	0.00589326	0.00294663	+	0.999996i	$0.499062\pi$
$594$	0	0
$595$	−820.669	−0.0565448
$596$	0	0
$597$	12504.5	0.857246
$598$	0	0
$599$	13447.6	0.917287	0.458643	−	0.888620i	$-0.348336\pi$
$599$	13447.6	0.917287	0.458643	+	0.888620i	$0.348336\pi$
$600$	0	0
$601$	9889.36	0.671207	0.335603	−	0.942003i	$-0.391060\pi$
$601$	9889.36	0.671207	0.335603	+	0.942003i	$0.391060\pi$
$602$	0	0
$603$	−5622.61	−0.379719
$604$	0	0
$605$	−9734.03	−0.654123
$606$	0	0
$607$	−24781.2	−1.65706	−0.828531	−	0.559943i	$-0.810823\pi$
$607$	−24781.2	−1.65706	−0.828531	+	0.559943i	$0.810823\pi$
$608$	0	0
$609$	−3055.12	−0.203284
$610$	0	0
$611$	−34838.4	−2.30673
$612$	0	0
$613$	−16199.4	−1.06736	−0.533678	−	0.845688i	$-0.679191\pi$
$613$	−16199.4	−1.06736	−0.533678	+	0.845688i	$0.679191\pi$
$614$	0	0
$615$	9288.69	0.609035
$616$	0	0
$617$	12299.5	0.802527	0.401263	−	0.915963i	$-0.368571\pi$
$617$	12299.5	0.802527	0.401263	+	0.915963i	$0.368571\pi$
$618$	0	0
$619$	22374.5	1.45284	0.726420	−	0.687251i	$-0.241183\pi$
$619$	22374.5	1.45284	0.726420	+	0.687251i	$0.241183\pi$
$620$	0	0
$621$	−2014.30	−0.130163
$622$	0	0
$623$	−2805.41	−0.180412
$624$	0	0
$625$	−17566.1	−1.12423
$626$	0	0
$627$	5473.93	0.348656
$628$	0	0
$629$	44.8880	0.00284547
$630$	0	0
$631$	8319.83	0.524892	0.262446	−	0.964947i	$-0.415471\pi$
$631$	8319.83	0.524892	0.262446	+	0.964947i	$0.415471\pi$
$632$	0	0
$633$	3944.19	0.247658
$634$	0	0
$635$	−36413.9	−2.27565
$636$	0	0
$637$	3617.65	0.225018
$638$	0	0
$639$	−1825.37	−0.113006
$640$	0	0
$641$	21003.0	1.29418	0.647088	−	0.762415i	$-0.275987\pi$
$641$	21003.0	1.29418	0.647088	+	0.762415i	$0.275987\pi$
$642$	0	0
$643$	2306.90	0.141486	0.0707428	−	0.997495i	$-0.477463\pi$
$643$	2306.90	0.141486	0.0707428	+	0.997495i	$0.477463\pi$
$644$	0	0
$645$	−4369.51	−0.266743
$646$	0	0
$647$	−14348.5	−0.871866	−0.435933	−	0.899979i	$-0.643582\pi$
$647$	−14348.5	−0.871866	−0.435933	+	0.899979i	$0.643582\pi$
$648$	0	0
$649$	21949.7	1.32758
$650$	0	0
$651$	1663.82	0.100169
$652$	0	0
$653$	−15852.0	−0.949982	−0.474991	−	0.879991i	$-0.657549\pi$
$653$	−15852.0	−0.949982	−0.474991	+	0.879991i	$0.657549\pi$
$654$	0	0
$655$	29294.8	1.74755
$656$	0	0
$657$	7477.93	0.444051
$658$	0	0
$659$	12605.9	0.745151	0.372576	−	0.928002i	$-0.378475\pi$
$659$	12605.9	0.745151	0.372576	+	0.928002i	$0.378475\pi$
$660$	0	0
$661$	−1234.33	−0.0726324	−0.0363162	−	0.999340i	$-0.511562\pi$
$661$	−1234.33	−0.0726324	−0.0363162	+	0.999340i	$0.511562\pi$
$662$	0	0
$663$	1705.44	0.0999004
$664$	0	0
$665$	−7394.40	−0.431192
$666$	0	0
$667$	−10853.5	−0.630059
$668$	0	0
$669$	−13597.3	−0.785804
$670$	0	0
$671$	−19312.1	−1.11108
$672$	0	0
$673$	2234.45	0.127982	0.0639909	−	0.997950i	$-0.479617\pi$
$673$	2234.45	0.127982	0.0639909	+	0.997950i	$0.479617\pi$
$674$	0	0
$675$	2884.40	0.164475
$676$	0	0
$677$	−17611.9	−0.999825	−0.499913	−	0.866076i	$-0.666635\pi$
$677$	−17611.9	−0.999825	−0.499913	+	0.866076i	$0.666635\pi$
$678$	0	0
$679$	−833.104	−0.0470863
$680$	0	0
$681$	12660.7	0.712423
$682$	0	0
$683$	−30074.3	−1.68486	−0.842431	−	0.538804i	$-0.818876\pi$
$683$	−30074.3	−1.68486	−0.842431	+	0.538804i	$0.818876\pi$
$684$	0	0
$685$	−18480.0	−1.03078
$686$	0	0
$687$	18388.8	1.02121
$688$	0	0
$689$	−26688.6	−1.47570
$690$	0	0
$691$	−13009.6	−0.716220	−0.358110	−	0.933679i	$-0.616579\pi$
$691$	−13009.6	−0.716220	−0.358110	+	0.933679i	$0.616579\pi$
$692$	0	0
$693$	−1656.91	−0.0908234
$694$	0	0
$695$	−2727.15	−0.148844
$696$	0	0
$697$	1565.79	0.0850913
$698$	0	0
$699$	11102.0	0.600740
$700$	0	0
$701$	−16387.7	−0.882961	−0.441481	−	0.897271i	$-0.645547\pi$
$701$	−16387.7	−0.882961	−0.441481	+	0.897271i	$0.645547\pi$
$702$	0	0
$703$	404.450	0.0216986
$704$	0	0
$705$	−21554.2	−1.15146
$706$	0	0
$707$	−5580.63	−0.296862
$708$	0	0
$709$	−2577.34	−0.136522	−0.0682610	−	0.997667i	$-0.521745\pi$
$709$	−2577.34	−0.136522	−0.0682610	+	0.997667i	$0.521745\pi$
$710$	0	0
$711$	7636.21	0.402785
$712$	0	0
$713$	5910.81	0.310465
$714$	0	0
$715$	29564.7	1.54637
$716$	0	0
$717$	−18879.0	−0.983333
$718$	0	0
$719$	−36033.1	−1.86900	−0.934498	−	0.355969i	$-0.884151\pi$
$719$	−36033.1	−1.86900	−0.934498	+	0.355969i	$0.884151\pi$
$720$	0	0
$721$	2225.60	0.114959
$722$	0	0
$723$	−5916.02	−0.304314
$724$	0	0
$725$	15541.8	0.796148
$726$	0	0
$727$	−17453.2	−0.890377	−0.445189	−	0.895437i	$-0.646863\pi$
$727$	−17453.2	−0.890377	−0.445189	+	0.895437i	$0.646863\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	−736.568	−0.0372681
$732$	0	0
$733$	−34145.8	−1.72060	−0.860302	−	0.509784i	$-0.829725\pi$
$733$	−34145.8	−1.72060	−0.860302	+	0.509784i	$0.829725\pi$
$734$	0	0
$735$	2238.22	0.112324
$736$	0	0
$737$	−16430.6	−0.821205
$738$	0	0
$739$	−33602.5	−1.67265	−0.836325	−	0.548234i	$-0.815300\pi$
$739$	−33602.5	−1.67265	−0.836325	+	0.548234i	$0.815300\pi$
$740$	0	0
$741$	15366.4	0.761807
$742$	0	0
$743$	13644.0	0.673687	0.336844	−	0.941561i	$-0.390641\pi$
$743$	13644.0	0.673687	0.336844	+	0.941561i	$0.390641\pi$
$744$	0	0
$745$	−9562.43	−0.470255
$746$	0	0
$747$	7004.67	0.343089
$748$	0	0
$749$	7008.79	0.341917
$750$	0	0
$751$	−25970.9	−1.26191	−0.630953	−	0.775821i	$-0.717336\pi$
$751$	−25970.9	−1.26191	−0.630953	+	0.775821i	$0.717336\pi$
$752$	0	0
$753$	−15988.0	−0.773753
$754$	0	0
$755$	13959.6	0.672902
$756$	0	0
$757$	2187.49	0.105027	0.0525137	−	0.998620i	$-0.483277\pi$
$757$	2187.49	0.105027	0.0525137	+	0.998620i	$0.483277\pi$
$758$	0	0
$759$	−5886.25	−0.281499
$760$	0	0
$761$	−12262.9	−0.584140	−0.292070	−	0.956397i	$-0.594344\pi$
$761$	−12262.9	−0.584140	−0.292070	+	0.956397i	$0.594344\pi$
$762$	0	0
$763$	−1969.23	−0.0934351
$764$	0	0
$765$	1055.15	0.0498678
$766$	0	0
$767$	61617.2	2.90074
$768$	0	0
$769$	−13427.8	−0.629672	−0.314836	−	0.949146i	$-0.601949\pi$
$769$	−13427.8	−0.629672	−0.314836	+	0.949146i	$0.601949\pi$
$770$	0	0
$771$	10026.7	0.468357
$772$	0	0
$773$	−26591.7	−1.23730	−0.618652	−	0.785665i	$-0.712321\pi$
$773$	−26591.7	−1.23730	−0.618652	+	0.785665i	$0.712321\pi$
$774$	0	0
$775$	−8464.06	−0.392307
$776$	0	0
$777$	−122.423	−0.00565239
$778$	0	0
$779$	14108.1	0.648878
$780$	0	0
$781$	−5334.16	−0.244394
$782$	0	0
$783$	3928.01	0.179279
$784$	0	0
$785$	−39773.0	−1.80835
$786$	0	0
$787$	8475.66	0.383894	0.191947	−	0.981405i	$-0.438520\pi$
$787$	8475.66	0.383894	0.191947	+	0.981405i	$0.438520\pi$
$788$	0	0
$789$	2986.22	0.134743
$790$	0	0
$791$	−9312.48	−0.418602
$792$	0	0
$793$	−54213.0	−2.42769
$794$	0	0
$795$	−16512.0	−0.736631
$796$	0	0
$797$	−23049.6	−1.02442	−0.512209	−	0.858861i	$-0.671173\pi$
$797$	−23049.6	−1.02442	−0.512209	+	0.858861i	$0.671173\pi$
$798$	0	0
$799$	−3633.39	−0.160876
$800$	0	0
$801$	3606.96	0.159108
$802$	0	0
$803$	21852.2	0.960335
$804$	0	0
$805$	7951.39	0.348136
$806$	0	0
$807$	18869.0	0.823073
$808$	0	0
$809$	−16971.3	−0.737551	−0.368776	−	0.929518i	$-0.620223\pi$
$809$	−16971.3	−0.737551	−0.368776	+	0.929518i	$0.620223\pi$
$810$	0	0
$811$	31319.6	1.35608	0.678038	−	0.735027i	$-0.262830\pi$
$811$	31319.6	1.35608	0.678038	+	0.735027i	$0.262830\pi$
$812$	0	0
$813$	22396.1	0.966131
$814$	0	0
$815$	−39660.7	−1.70461
$816$	0	0
$817$	−6636.63	−0.284194
$818$	0	0
$819$	−4651.27	−0.198448
$820$	0	0
$821$	35671.3	1.51637	0.758184	−	0.652041i	$-0.226087\pi$
$821$	35671.3	1.51637	0.758184	+	0.652041i	$0.226087\pi$
$822$	0	0
$823$	26077.5	1.10450	0.552250	−	0.833678i	$-0.313769\pi$
$823$	26077.5	1.10450	0.552250	+	0.833678i	$0.313769\pi$
$824$	0	0
$825$	8428.89	0.355705
$826$	0	0
$827$	30300.3	1.27406	0.637028	−	0.770841i	$-0.280164\pi$
$827$	30300.3	1.27406	0.637028	+	0.770841i	$0.280164\pi$
$828$	0	0
$829$	6276.23	0.262946	0.131473	−	0.991320i	$-0.458029\pi$
$829$	6276.23	0.262946	0.131473	+	0.991320i	$0.458029\pi$
$830$	0	0
$831$	12827.8	0.535491
$832$	0	0
$833$	377.295	0.0156933
$834$	0	0
$835$	−33835.1	−1.40229
$836$	0	0
$837$	−2139.20	−0.0883410
$838$	0	0
$839$	−5255.20	−0.216245	−0.108123	−	0.994138i	$-0.534484\pi$
$839$	−5255.20	−0.216245	−0.108123	+	0.994138i	$0.534484\pi$
$840$	0	0
$841$	−3224.00	−0.132191
$842$	0	0
$843$	−8644.01	−0.353162
$844$	0	0
$845$	49542.5	2.01694
$846$	0	0
$847$	4475.14	0.181544
$848$	0	0
$849$	20441.9	0.826340
$850$	0	0
$851$	−434.916	−0.0175191
$852$	0	0
$853$	7945.91	0.318948	0.159474	−	0.987202i	$-0.449020\pi$
$853$	7945.91	0.318948	0.159474	+	0.987202i	$0.449020\pi$
$854$	0	0
$855$	9507.08	0.380275
$856$	0	0
$857$	−26884.5	−1.07159	−0.535797	−	0.844347i	$-0.679989\pi$
$857$	−26884.5	−1.07159	−0.535797	+	0.844347i	$0.679989\pi$
$858$	0	0
$859$	6748.79	0.268063	0.134031	−	0.990977i	$-0.457208\pi$
$859$	6748.79	0.268063	0.134031	+	0.990977i	$0.457208\pi$
$860$	0	0
$861$	−4270.40	−0.169030
$862$	0	0
$863$	11974.3	0.472319	0.236159	−	0.971714i	$-0.424111\pi$
$863$	11974.3	0.472319	0.236159	+	0.971714i	$0.424111\pi$
$864$	0	0
$865$	−12982.0	−0.510289
$866$	0	0
$867$	−14561.1	−0.570383
$868$	0	0
$869$	22314.8	0.871090
$870$	0	0
$871$	−46123.9	−1.79432
$872$	0	0
$873$	1071.13	0.0415262
$874$	0	0
$875$	1936.62	0.0748227
$876$	0	0
$877$	19731.1	0.759717	0.379859	−	0.925045i	$-0.375973\pi$
$877$	19731.1	0.759717	0.379859	+	0.925045i	$0.375973\pi$
$878$	0	0
$879$	24724.7	0.948740
$880$	0	0
$881$	13625.5	0.521059	0.260530	−	0.965466i	$-0.416103\pi$
$881$	13625.5	0.521059	0.260530	+	0.965466i	$0.416103\pi$
$882$	0	0
$883$	38311.5	1.46012	0.730059	−	0.683384i	$-0.239492\pi$
$883$	38311.5	1.46012	0.730059	+	0.683384i	$0.239492\pi$
$884$	0	0
$885$	38122.1	1.44798
$886$	0	0
$887$	−24106.5	−0.912534	−0.456267	−	0.889843i	$-0.650814\pi$
$887$	−24106.5	−0.912534	−0.456267	+	0.889843i	$0.650814\pi$
$888$	0	0
$889$	16741.0	0.631579
$890$	0	0
$891$	2130.31	0.0800988
$892$	0	0
$893$	−32737.6	−1.22679
$894$	0	0
$895$	−32021.4	−1.19593
$896$	0	0
$897$	−16523.9	−0.615069
$898$	0	0
$899$	−11526.5	−0.427618
$900$	0	0
$901$	−2783.43	−0.102918
$902$	0	0
$903$	2008.85	0.0740313
$904$	0	0
$905$	70570.1	2.59208
$906$	0	0
$907$	43386.9	1.58836	0.794178	−	0.607685i	$-0.207902\pi$
$907$	43386.9	1.58836	0.794178	+	0.607685i	$0.207902\pi$
$908$	0	0
$909$	7175.10	0.261808
$910$	0	0
$911$	19774.0	0.719145	0.359573	−	0.933117i	$-0.382922\pi$
$911$	19774.0	0.719145	0.359573	+	0.933117i	$0.382922\pi$
$912$	0	0
$913$	20469.3	0.741987
$914$	0	0
$915$	−33541.1	−1.21184
$916$	0	0
$917$	−13468.0	−0.485009
$918$	0	0
$919$	3437.54	0.123388	0.0616942	−	0.998095i	$-0.480350\pi$
$919$	3437.54	0.123388	0.0616942	+	0.998095i	$0.480350\pi$
$920$	0	0
$921$	−20455.8	−0.731858
$922$	0	0
$923$	−14974.1	−0.533995
$924$	0	0
$925$	622.783	0.0221373
$926$	0	0
$927$	−2861.49	−0.101385
$928$	0	0
$929$	−16677.2	−0.588979	−0.294490	−	0.955655i	$-0.595150\pi$
$929$	−16677.2	−0.588979	−0.294490	+	0.955655i	$0.595150\pi$
$930$	0	0
$931$	3399.51	0.119672
$932$	0	0
$933$	−26626.0	−0.934295
$934$	0	0
$935$	3083.38	0.107847
$936$	0	0
$937$	15356.2	0.535396	0.267698	−	0.963503i	$-0.413737\pi$
$937$	15356.2	0.535396	0.267698	+	0.963503i	$0.413737\pi$
$938$	0	0
$939$	16762.1	0.582546
$940$	0	0
$941$	−22571.5	−0.781944	−0.390972	−	0.920403i	$-0.627861\pi$
$941$	−22571.5	−0.781944	−0.390972	+	0.920403i	$0.627861\pi$
$942$	0	0
$943$	−15170.8	−0.523892
$944$	0	0
$945$	−2877.71	−0.0990601
$946$	0	0
$947$	−19792.6	−0.679169	−0.339585	−	0.940575i	$-0.610287\pi$
$947$	−19792.6	−0.679169	−0.339585	+	0.940575i	$0.610287\pi$
$948$	0	0
$949$	61343.7	2.09831
$950$	0	0
$951$	−15555.3	−0.530404
$952$	0	0
$953$	24230.5	0.823614	0.411807	−	0.911271i	$-0.364898\pi$
$953$	24230.5	0.823614	0.411807	+	0.911271i	$0.364898\pi$
$954$	0	0
$955$	−9142.58	−0.309787
$956$	0	0
$957$	11478.6	0.387721
$958$	0	0
$959$	8496.00	0.286079
$960$	0	0
$961$	−23513.7	−0.789288
$962$	0	0
$963$	−9011.30	−0.301542
$964$	0	0
$965$	48122.3	1.60530
$966$	0	0
$967$	−2679.98	−0.0891234	−0.0445617	−	0.999007i	$-0.514189\pi$
$967$	−2679.98	−0.0891234	−0.0445617	+	0.999007i	$0.514189\pi$
$968$	0	0
$969$	1602.61	0.0531302
$970$	0	0
$971$	−54651.0	−1.80621	−0.903107	−	0.429416i	$-0.858719\pi$
$971$	−54651.0	−1.80621	−0.903107	+	0.429416i	$0.858719\pi$
$972$	0	0
$973$	1253.78	0.0413098
$974$	0	0
$975$	23661.6	0.777208
$976$	0	0
$977$	7678.44	0.251438	0.125719	−	0.992066i	$-0.459876\pi$
$977$	7678.44	0.251438	0.125719	+	0.992066i	$0.459876\pi$
$978$	0	0
$979$	10540.4	0.344098
$980$	0	0
$981$	2531.87	0.0824020
$982$	0	0
$983$	39582.6	1.28432	0.642162	−	0.766569i	$-0.278038\pi$
$983$	39582.6	1.28432	0.642162	+	0.766569i	$0.278038\pi$
$984$	0	0
$985$	60694.1	1.96333
$986$	0	0
$987$	9909.37	0.319573
$988$	0	0
$989$	7136.54	0.229453
$990$	0	0
$991$	32079.0	1.02828	0.514139	−	0.857707i	$-0.328112\pi$
$991$	32079.0	1.02828	0.514139	+	0.857707i	$0.328112\pi$
$992$	0	0
$993$	28973.1	0.925915
$994$	0	0
$995$	63464.4	2.02207
$996$	0	0
$997$	58882.0	1.87042	0.935212	−	0.354089i	$-0.115209\pi$
$997$	58882.0	1.87042	0.935212	+	0.354089i	$0.115209\pi$
$998$	0	0
$999$	157.401	0.00498494

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1344.4.a.bu.1.3		3
4.3	odd	2		1344.4.a.bs.1.3		3
8.3	odd	2		672.4.a.r.1.1	yes	3
8.5	even	2		672.4.a.p.1.1	✓	3
24.5	odd	2		2016.4.a.u.1.3		3
24.11	even	2		2016.4.a.v.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
672.4.a.p.1.1	✓	3	8.5	even	2
672.4.a.r.1.1	yes	3	8.3	odd	2
1344.4.a.bs.1.3		3	4.3	odd	2
1344.4.a.bu.1.3		3	1.1	even	1	trivial
2016.4.a.u.1.3		3	24.5	odd	2
2016.4.a.v.1.3		3	24.11	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1344 = 2^{6} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1344.a (trivial)

\(f(q)\)	\(=\)	\(q+3.00000 q^{3} +15.2260 q^{5} -7.00000 q^{7} +9.00000 q^{9} +O(q^{10})\)	\(q+3.00000 q^{3} +15.2260 q^{5} -7.00000 q^{7} +9.00000 q^{9} +26.3001 q^{11} +73.8297 q^{13} +45.6779 q^{15} +7.69991 q^{17} +69.3778 q^{19} -21.0000 q^{21} -74.6037 q^{23} +106.830 q^{25} +27.0000 q^{27} +145.482 q^{29} -79.2295 q^{31} +78.9003 q^{33} -106.582 q^{35} +5.82968 q^{37} +221.489 q^{39} +203.352 q^{41} -95.6594 q^{43} +137.034 q^{45} -471.875 q^{47} +49.0000 q^{49} +23.0997 q^{51} -361.489 q^{53} +400.444 q^{55} +208.133 q^{57} +834.586 q^{59} -734.298 q^{61} -63.0000 q^{63} +1124.13 q^{65} -624.735 q^{67} -223.811 q^{69} -202.819 q^{71} +830.881 q^{73} +320.489 q^{75} -184.101 q^{77} +848.468 q^{79} +81.0000 q^{81} +778.297 q^{83} +117.238 q^{85} +436.446 q^{87} +400.773 q^{89} -516.808 q^{91} -237.688 q^{93} +1056.34 q^{95} +119.015 q^{97} +236.701 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 9 q^{3} - 6 q^{5} - 21 q^{7} + 27 q^{9}+O(q^{10}) \) 3 * q + 9 * q^3 - 6 * q^5 - 21 * q^7 + 27 * q^9	\( 3 q + 9 q^{3} - 6 q^{5} - 21 q^{7} + 27 q^{9} + 48 q^{11} - 6 q^{13} - 18 q^{15} + 54 q^{17} + 84 q^{19} - 63 q^{21} - 48 q^{23} + 93 q^{25} + 81 q^{27} - 18 q^{29} - 72 q^{31} + 144 q^{33} + 42 q^{35} - 210 q^{37} - 18 q^{39} + 414 q^{41} + 168 q^{43} - 54 q^{45} - 72 q^{47} + 147 q^{49} + 162 q^{51} - 402 q^{53} + 456 q^{55} + 252 q^{57} + 540 q^{59} - 798 q^{61} - 189 q^{63} + 1740 q^{65} + 48 q^{67} - 144 q^{69} + 456 q^{71} + 1230 q^{73} + 279 q^{75} - 336 q^{77} + 1368 q^{79} + 243 q^{81} + 60 q^{83} - 660 q^{85} - 54 q^{87} + 2742 q^{89} + 42 q^{91} - 216 q^{93} + 648 q^{95} + 1950 q^{97} + 432 q^{99}+O(q^{100}) \) 3 * q + 9 * q^3 - 6 * q^5 - 21 * q^7 + 27 * q^9 + 48 * q^11 - 6 * q^13 - 18 * q^15 + 54 * q^17 + 84 * q^19 - 63 * q^21 - 48 * q^23 + 93 * q^25 + 81 * q^27 - 18 * q^29 - 72 * q^31 + 144 * q^33 + 42 * q^35 - 210 * q^37 - 18 * q^39 + 414 * q^41 + 168 * q^43 - 54 * q^45 - 72 * q^47 + 147 * q^49 + 162 * q^51 - 402 * q^53 + 456 * q^55 + 252 * q^57 + 540 * q^59 - 798 * q^61 - 189 * q^63 + 1740 * q^65 + 48 * q^67 - 144 * q^69 + 456 * q^71 + 1230 * q^73 + 279 * q^75 - 336 * q^77 + 1368 * q^79 + 243 * q^81 + 60 * q^83 - 660 * q^85 - 54 * q^87 + 2742 * q^89 + 42 * q^91 - 216 * q^93 + 648 * q^95 + 1950 * q^97 + 432 * q^99

Introduction

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