LMFDB - Embedded newform 1344.4.a.bn.1.1

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:	$2$
Coefficient field:	$\Q(\sqrt{11}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 11 $ x^2 - 11
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 672)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-3.31662$ 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)

$f(q)$	$=$	$q+3.00000 q^{3} -10.6332 q^{5} +7.00000 q^{7} +9.00000 q^{9} +O(q^{10})$	$q+3.00000 q^{3} -10.6332 q^{5} +7.00000 q^{7} +9.00000 q^{9} -68.4327 q^{11} -23.0660 q^{13} -31.8997 q^{15} -62.2322 q^{17} -53.2665 q^{19} +21.0000 q^{21} +89.8997 q^{23} -11.9340 q^{25} +27.0000 q^{27} +43.2665 q^{29} +102.734 q^{31} -205.298 q^{33} -74.4327 q^{35} +302.264 q^{37} -69.1980 q^{39} +73.4302 q^{41} +377.330 q^{43} -95.6992 q^{45} +487.662 q^{47} +49.0000 q^{49} -186.697 q^{51} -467.594 q^{53} +727.662 q^{55} -159.799 q^{57} +432.327 q^{59} +70.4645 q^{61} +63.0000 q^{63} +245.266 q^{65} -475.799 q^{67} +269.699 q^{69} -680.824 q^{71} +604.069 q^{73} -35.8020 q^{75} -479.029 q^{77} +329.140 q^{79} +81.0000 q^{81} -834.670 q^{83} +661.731 q^{85} +129.799 q^{87} -947.890 q^{89} -161.462 q^{91} +308.201 q^{93} +566.396 q^{95} +661.272 q^{97} -615.895 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 6 q^{3} - 8 q^{5} + 14 q^{7} + 18 q^{9}+O(q^{10}) $ 2 * q + 6 * q^3 - 8 * q^5 + 14 * q^7 + 18 * q^9	$ 2 q + 6 q^{3} - 8 q^{5} + 14 q^{7} + 18 q^{9} - 44 q^{11} + 60 q^{13} - 24 q^{15} + 48 q^{17} - 80 q^{19} + 42 q^{21} + 140 q^{23} - 130 q^{25} + 54 q^{27} + 60 q^{29} + 232 q^{31} - 132 q^{33} - 56 q^{35} + 180 q^{37} + 180 q^{39} - 344 q^{41} + 224 q^{43} - 72 q^{45} + 312 q^{47} + 98 q^{49} + 144 q^{51} + 20 q^{53} + 792 q^{55} - 240 q^{57} - 64 q^{59} - 204 q^{61} + 126 q^{63} + 464 q^{65} - 872 q^{67} + 420 q^{69} + 164 q^{71} + 1500 q^{73} - 390 q^{75} - 308 q^{77} + 1640 q^{79} + 162 q^{81} - 2200 q^{83} + 952 q^{85} + 180 q^{87} - 264 q^{89} + 420 q^{91} + 696 q^{93} + 496 q^{95} + 2092 q^{97} - 396 q^{99}+O(q^{100}) $ 2 * q + 6 * q^3 - 8 * q^5 + 14 * q^7 + 18 * q^9 - 44 * q^11 + 60 * q^13 - 24 * q^15 + 48 * q^17 - 80 * q^19 + 42 * q^21 + 140 * q^23 - 130 * q^25 + 54 * q^27 + 60 * q^29 + 232 * q^31 - 132 * q^33 - 56 * q^35 + 180 * q^37 + 180 * q^39 - 344 * q^41 + 224 * q^43 - 72 * q^45 + 312 * q^47 + 98 * q^49 + 144 * q^51 + 20 * q^53 + 792 * q^55 - 240 * q^57 - 64 * q^59 - 204 * q^61 + 126 * q^63 + 464 * q^65 - 872 * q^67 + 420 * q^69 + 164 * q^71 + 1500 * q^73 - 390 * q^75 - 308 * q^77 + 1640 * q^79 + 162 * q^81 - 2200 * q^83 + 952 * q^85 + 180 * q^87 - 264 * q^89 + 420 * q^91 + 696 * q^93 + 496 * q^95 + 2092 * q^97 - 396 * 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
$5$	−10.6332	−0.951067	−0.475533	−	0.879698i	$-0.657745\pi$
$5$	−10.6332	−0.951067	−0.475533	+	0.879698i	$0.657745\pi$
$6$	0	0
$7$	7.00000	0.377964
$7$	7.00000	0.377964
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	−68.4327	−1.87575	−0.937875	−	0.346973i	$-0.887209\pi$
$11$	−68.4327	−1.87575	−0.937875	+	0.346973i	$0.887209\pi$
$12$	0	0
$13$	−23.0660	−0.492104	−0.246052	−	0.969257i	$-0.579133\pi$
$13$	−23.0660	−0.492104	−0.246052	+	0.969257i	$0.579133\pi$
$14$	0	0
$15$	−31.8997	−0.549099
$16$	0	0
$17$	−62.2322	−0.887855	−0.443928	−	0.896063i	$-0.646415\pi$
$17$	−62.2322	−0.887855	−0.443928	+	0.896063i	$0.646415\pi$
$18$	0	0
$19$	−53.2665	−0.643167	−0.321583	−	0.946881i	$-0.604215\pi$
$19$	−53.2665	−0.643167	−0.321583	+	0.946881i	$0.604215\pi$
$20$	0	0
$21$	21.0000	0.218218
$22$	0	0
$23$	89.8997	0.815017	0.407509	−	0.913201i	$-0.366398\pi$
$23$	89.8997	0.815017	0.407509	+	0.913201i	$0.366398\pi$
$24$	0	0
$25$	−11.9340	−0.0954720
$26$	0	0
$27$	27.0000	0.192450
$28$	0	0
$29$	43.2665	0.277048	0.138524	−	0.990359i	$-0.455764\pi$
$29$	43.2665	0.277048	0.138524	+	0.990359i	$0.455764\pi$
$30$	0	0
$31$	102.734	0.595209	0.297605	−	0.954689i	$-0.403812\pi$
$31$	102.734	0.595209	0.297605	+	0.954689i	$0.403812\pi$
$32$	0	0
$33$	−205.298	−1.08296
$34$	0	0
$35$	−74.4327	−0.359469
$36$	0	0
$37$	302.264	1.34302	0.671512	−	0.740994i	$-0.265645\pi$
$37$	302.264	1.34302	0.671512	+	0.740994i	$0.265645\pi$
$38$	0	0
$39$	−69.1980	−0.284117
$40$	0	0
$41$	73.4302	0.279704	0.139852	−	0.990172i	$-0.455337\pi$
$41$	73.4302	0.279704	0.139852	+	0.990172i	$0.455337\pi$
$42$	0	0
$43$	377.330	1.33819	0.669096	−	0.743176i	$-0.266681\pi$
$43$	377.330	1.33819	0.669096	+	0.743176i	$0.266681\pi$
$44$	0	0
$45$	−95.6992	−0.317022
$46$	0	0
$47$	487.662	1.51347	0.756733	−	0.653724i	$-0.226794\pi$
$47$	487.662	1.51347	0.756733	+	0.653724i	$0.226794\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	0	0
$51$	−186.697	−0.512603
$52$	0	0
$53$	−467.594	−1.21187	−0.605934	−	0.795515i	$-0.707200\pi$
$53$	−467.594	−1.21187	−0.605934	+	0.795515i	$0.707200\pi$
$54$	0	0
$55$	727.662	1.78396
$56$	0	0
$57$	−159.799	−0.371333
$58$	0	0
$59$	432.327	0.953970	0.476985	−	0.878911i	$-0.341730\pi$
$59$	432.327	0.953970	0.476985	+	0.878911i	$0.341730\pi$
$60$	0	0
$61$	70.4645	0.147903	0.0739513	−	0.997262i	$-0.476439\pi$
$61$	70.4645	0.147903	0.0739513	+	0.997262i	$0.476439\pi$
$62$	0	0
$63$	63.0000	0.125988
$64$	0	0
$65$	245.266	0.468024
$66$	0	0
$67$	−475.799	−0.867584	−0.433792	−	0.901013i	$-0.642825\pi$
$67$	−475.799	−0.867584	−0.433792	+	0.901013i	$0.642825\pi$
$68$	0	0
$69$	269.699	0.470550
$70$	0	0
$71$	−680.824	−1.13801	−0.569006	−	0.822333i	$-0.692672\pi$
$71$	−680.824	−1.13801	−0.569006	+	0.822333i	$0.692672\pi$
$72$	0	0
$73$	604.069	0.968505	0.484253	−	0.874928i	$-0.339092\pi$
$73$	604.069	0.968505	0.484253	+	0.874928i	$0.339092\pi$
$74$	0	0
$75$	−35.8020	−0.0551208
$76$	0	0
$77$	−479.029	−0.708967
$78$	0	0
$79$	329.140	0.468748	0.234374	−	0.972147i	$-0.424696\pi$
$79$	329.140	0.468748	0.234374	+	0.972147i	$0.424696\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	−834.670	−1.10382	−0.551909	−	0.833904i	$-0.686101\pi$
$83$	−834.670	−1.10382	−0.551909	+	0.833904i	$0.686101\pi$
$84$	0	0
$85$	661.731	0.844409
$86$	0	0
$87$	129.799	0.159954
$88$	0	0
$89$	−947.890	−1.12895	−0.564473	−	0.825452i	$-0.690920\pi$
$89$	−947.890	−1.12895	−0.564473	+	0.825452i	$0.690920\pi$
$90$	0	0
$91$	−161.462	−0.185998
$92$	0	0
$93$	308.201	0.343644
$94$	0	0
$95$	566.396	0.611695
$96$	0	0
$97$	661.272	0.692185	0.346093	−	0.938200i	$-0.387508\pi$
$97$	661.272	0.692185	0.346093	+	0.938200i	$0.387508\pi$
$98$	0	0
$99$	−615.895	−0.625250
$100$	0	0
$101$	745.293	0.734252	0.367126	−	0.930171i	$-0.380342\pi$
$101$	745.293	0.734252	0.367126	+	0.930171i	$0.380342\pi$
$102$	0	0
$103$	863.261	0.825822	0.412911	−	0.910771i	$-0.364512\pi$
$103$	863.261	0.825822	0.412911	+	0.910771i	$0.364512\pi$
$104$	0	0
$105$	−223.298	−0.207540
$106$	0	0
$107$	−1198.42	−1.08276	−0.541380	−	0.840778i	$-0.682098\pi$
$107$	−1198.42	−1.08276	−0.541380	+	0.840778i	$0.682098\pi$
$108$	0	0
$109$	84.1320	0.0739301	0.0369651	−	0.999317i	$-0.488231\pi$
$109$	84.1320	0.0739301	0.0369651	+	0.999317i	$0.488231\pi$
$110$	0	0
$111$	906.792	0.775395
$112$	0	0
$113$	2030.12	1.69006	0.845032	−	0.534715i	$-0.179581\pi$
$113$	2030.12	1.69006	0.845032	+	0.534715i	$0.179581\pi$
$114$	0	0
$115$	−955.926	−0.775136
$116$	0	0
$117$	−207.594	−0.164035
$118$	0	0
$119$	−435.626	−0.335578
$120$	0	0
$121$	3352.04	2.51844
$122$	0	0
$123$	220.291	0.161487
$124$	0	0
$125$	1456.05	1.04187
$126$	0	0
$127$	1363.12	0.952423	0.476212	−	0.879331i	$-0.342010\pi$
$127$	1363.12	0.952423	0.476212	+	0.879331i	$0.342010\pi$
$128$	0	0
$129$	1131.99	0.772606
$130$	0	0
$131$	−1115.20	−0.743781	−0.371890	−	0.928277i	$-0.621290\pi$
$131$	−1115.20	−0.743781	−0.371890	+	0.928277i	$0.621290\pi$
$132$	0	0
$133$	−372.865	−0.243094
$134$	0	0
$135$	−287.098	−0.183033
$136$	0	0
$137$	361.536	0.225460	0.112730	−	0.993626i	$-0.464040\pi$
$137$	361.536	0.225460	0.112730	+	0.993626i	$0.464040\pi$
$138$	0	0
$139$	−426.543	−0.260280	−0.130140	−	0.991496i	$-0.541543\pi$
$139$	−426.543	−0.260280	−0.130140	+	0.991496i	$0.541543\pi$
$140$	0	0
$141$	1462.99	0.873800
$142$	0	0
$143$	1578.47	0.923065
$144$	0	0
$145$	−460.063	−0.263491
$146$	0	0
$147$	147.000	0.0824786
$148$	0	0
$149$	2861.46	1.57329	0.786645	−	0.617406i	$-0.211816\pi$
$149$	2861.46	1.57329	0.786645	+	0.617406i	$0.211816\pi$
$150$	0	0
$151$	−816.127	−0.439838	−0.219919	−	0.975518i	$-0.570579\pi$
$151$	−816.127	−0.439838	−0.219919	+	0.975518i	$0.570579\pi$
$152$	0	0
$153$	−560.090	−0.295952
$154$	0	0
$155$	−1092.39	−0.566084
$156$	0	0
$157$	−3243.63	−1.64885	−0.824426	−	0.565970i	$-0.808502\pi$
$157$	−3243.63	−1.64885	−0.824426	+	0.565970i	$0.808502\pi$
$158$	0	0
$159$	−1402.78	−0.699672
$160$	0	0
$161$	629.298	0.308048
$162$	0	0
$163$	3363.76	1.61638	0.808190	−	0.588922i	$-0.200447\pi$
$163$	3363.76	1.61638	0.808190	+	0.588922i	$0.200447\pi$
$164$	0	0
$165$	2182.99	1.02997
$166$	0	0
$167$	1352.59	0.626747	0.313373	−	0.949630i	$-0.398541\pi$
$167$	1352.59	0.626747	0.313373	+	0.949630i	$0.398541\pi$
$168$	0	0
$169$	−1664.96	−0.757833
$170$	0	0
$171$	−479.398	−0.214389
$172$	0	0
$173$	3491.30	1.53433	0.767163	−	0.641452i	$-0.221668\pi$
$173$	3491.30	1.53433	0.767163	+	0.641452i	$0.221668\pi$
$174$	0	0
$175$	−83.5380	−0.0360850
$176$	0	0
$177$	1296.98	0.550775
$178$	0	0
$179$	−2547.82	−1.06387	−0.531936	−	0.846785i	$-0.678535\pi$
$179$	−2547.82	−1.06387	−0.531936	+	0.846785i	$0.678535\pi$
$180$	0	0
$181$	358.254	0.147120	0.0735602	−	0.997291i	$-0.476564\pi$
$181$	358.254	0.147120	0.0735602	+	0.997291i	$0.476564\pi$
$182$	0	0
$183$	211.393	0.0853916
$184$	0	0
$185$	−3214.05	−1.27731
$186$	0	0
$187$	4258.72	1.66539
$188$	0	0
$189$	189.000	0.0727393
$190$	0	0
$191$	−2139.78	−0.810622	−0.405311	−	0.914179i	$-0.632837\pi$
$191$	−2139.78	−0.810622	−0.405311	+	0.914179i	$0.632837\pi$
$192$	0	0
$193$	2183.47	0.814349	0.407175	−	0.913350i	$-0.366514\pi$
$193$	2183.47	0.814349	0.407175	+	0.913350i	$0.366514\pi$
$194$	0	0
$195$	735.799	0.270214
$196$	0	0
$197$	1536.25	0.555600	0.277800	−	0.960639i	$-0.410395\pi$
$197$	1536.25	0.555600	0.277800	+	0.960639i	$0.410395\pi$
$198$	0	0
$199$	3413.15	1.21584	0.607920	−	0.793999i	$-0.292004\pi$
$199$	3413.15	1.21584	0.607920	+	0.793999i	$0.292004\pi$
$200$	0	0
$201$	−1427.40	−0.500900
$202$	0	0
$203$	302.865	0.104714
$204$	0	0
$205$	−780.802	−0.266017
$206$	0	0
$207$	809.098	0.271672
$208$	0	0
$209$	3645.17	1.20642
$210$	0	0
$211$	−2045.57	−0.667408	−0.333704	−	0.942678i	$-0.608299\pi$
$211$	−2045.57	−0.667408	−0.333704	+	0.942678i	$0.608299\pi$
$212$	0	0
$213$	−2042.47	−0.657032
$214$	0	0
$215$	−4012.24	−1.27271
$216$	0	0
$217$	719.135	0.224968
$218$	0	0
$219$	1812.21	0.559167
$220$	0	0
$221$	1435.45	0.436917
$222$	0	0
$223$	−4769.00	−1.43209	−0.716044	−	0.698055i	$-0.754049\pi$
$223$	−4769.00	−1.43209	−0.716044	+	0.698055i	$0.754049\pi$
$224$	0	0
$225$	−107.406	−0.0318240
$226$	0	0
$227$	3817.49	1.11619	0.558096	−	0.829776i	$-0.311532\pi$
$227$	3817.49	1.11619	0.558096	+	0.829776i	$0.311532\pi$
$228$	0	0
$229$	−288.289	−0.0831907	−0.0415954	−	0.999135i	$-0.513244\pi$
$229$	−288.289	−0.0831907	−0.0415954	+	0.999135i	$0.513244\pi$
$230$	0	0
$231$	−1437.09	−0.409322
$232$	0	0
$233$	9.15461	0.00257398	0.00128699	−	0.999999i	$-0.499590\pi$
$233$	9.15461	0.00257398	0.00128699	+	0.999999i	$0.499590\pi$
$234$	0	0
$235$	−5185.44	−1.43941
$236$	0	0
$237$	987.419	0.270632
$238$	0	0
$239$	−6630.35	−1.79449	−0.897243	−	0.441538i	$-0.854433\pi$
$239$	−6630.35	−1.79449	−0.897243	+	0.441538i	$0.854433\pi$
$240$	0	0
$241$	6727.26	1.79809	0.899047	−	0.437852i	$-0.144261\pi$
$241$	6727.26	1.79809	0.899047	+	0.437852i	$0.144261\pi$
$242$	0	0
$243$	243.000	0.0641500
$244$	0	0
$245$	−521.029	−0.135867
$246$	0	0
$247$	1228.64	0.316505
$248$	0	0
$249$	−2504.01	−0.637290
$250$	0	0
$251$	−877.794	−0.220741	−0.110370	−	0.993891i	$-0.535204\pi$
$251$	−877.794	−0.220741	−0.110370	+	0.993891i	$0.535204\pi$
$252$	0	0
$253$	−6152.09	−1.52877
$254$	0	0
$255$	1985.19	0.487520
$256$	0	0
$257$	−6332.65	−1.53704	−0.768521	−	0.639824i	$-0.779007\pi$
$257$	−6332.65	−1.53704	−0.768521	+	0.639824i	$0.779007\pi$
$258$	0	0
$259$	2115.85	0.507615
$260$	0	0
$261$	389.398	0.0923493
$262$	0	0
$263$	−3614.23	−0.847387	−0.423694	−	0.905806i	$-0.639267\pi$
$263$	−3614.23	−0.847387	−0.423694	+	0.905806i	$0.639267\pi$
$264$	0	0
$265$	4972.04	1.15257
$266$	0	0
$267$	−2843.67	−0.651797
$268$	0	0
$269$	7114.90	1.61265	0.806326	−	0.591472i	$-0.201453\pi$
$269$	7114.90	1.61265	0.806326	+	0.591472i	$0.201453\pi$
$270$	0	0
$271$	547.378	0.122697	0.0613485	−	0.998116i	$-0.480460\pi$
$271$	547.378	0.122697	0.0613485	+	0.998116i	$0.480460\pi$
$272$	0	0
$273$	−484.386	−0.107386
$274$	0	0
$275$	816.677	0.179082
$276$	0	0
$277$	5726.12	1.24205	0.621027	−	0.783789i	$-0.286716\pi$
$277$	5726.12	1.24205	0.621027	+	0.783789i	$0.286716\pi$
$278$	0	0
$279$	924.602	0.198403
$280$	0	0
$281$	237.378	0.0503943	0.0251972	−	0.999683i	$-0.491979\pi$
$281$	237.378	0.0503943	0.0251972	+	0.999683i	$0.491979\pi$
$282$	0	0
$283$	3179.66	0.667885	0.333942	−	0.942594i	$-0.391621\pi$
$283$	3179.66	0.667885	0.333942	+	0.942594i	$0.391621\pi$
$284$	0	0
$285$	1699.19	0.353162
$286$	0	0
$287$	514.012	0.105718
$288$	0	0
$289$	−1040.15	−0.211713
$290$	0	0
$291$	1983.81	0.399633
$292$	0	0
$293$	−7757.59	−1.54677	−0.773384	−	0.633937i	$-0.781438\pi$
$293$	−7757.59	−1.54677	−0.773384	+	0.633937i	$0.781438\pi$
$294$	0	0
$295$	−4597.05	−0.907289
$296$	0	0
$297$	−1847.68	−0.360988
$298$	0	0
$299$	−2073.63	−0.401073
$300$	0	0
$301$	2641.31	0.505789
$302$	0	0
$303$	2235.88	0.423921
$304$	0	0
$305$	−749.266	−0.140665
$306$	0	0
$307$	−8587.01	−1.59637	−0.798187	−	0.602410i	$-0.794207\pi$
$307$	−8587.01	−1.59637	−0.798187	+	0.602410i	$0.794207\pi$
$308$	0	0
$309$	2589.78	0.476789
$310$	0	0
$311$	−3889.43	−0.709163	−0.354581	−	0.935025i	$-0.615377\pi$
$311$	−3889.43	−0.709163	−0.354581	+	0.935025i	$0.615377\pi$
$312$	0	0
$313$	8408.84	1.51852	0.759259	−	0.650789i	$-0.225562\pi$
$313$	8408.84	1.51852	0.759259	+	0.650789i	$0.225562\pi$
$314$	0	0
$315$	−669.895	−0.119823
$316$	0	0
$317$	5651.73	1.00137	0.500683	−	0.865631i	$-0.333082\pi$
$317$	5651.73	1.00137	0.500683	+	0.865631i	$0.333082\pi$
$318$	0	0
$319$	−2960.85	−0.519672
$320$	0	0
$321$	−3595.25	−0.625132
$322$	0	0
$323$	3314.89	0.571039
$324$	0	0
$325$	275.270	0.0469822
$326$	0	0
$327$	252.396	0.0426836
$328$	0	0
$329$	3413.64	0.572036
$330$	0	0
$331$	7993.33	1.32735	0.663675	−	0.748021i	$-0.268996\pi$
$331$	7993.33	1.32735	0.663675	+	0.748021i	$0.268996\pi$
$332$	0	0
$333$	2720.38	0.447675
$334$	0	0
$335$	5059.29	0.825131
$336$	0	0
$337$	8193.32	1.32439	0.662194	−	0.749332i	$-0.269625\pi$
$337$	8193.32	1.32439	0.662194	+	0.749332i	$0.269625\pi$
$338$	0	0
$339$	6090.35	0.975759
$340$	0	0
$341$	−7030.34	−1.11646
$342$	0	0
$343$	343.000	0.0539949
$344$	0	0
$345$	−2867.78	−0.447525
$346$	0	0
$347$	8551.14	1.32291	0.661454	−	0.749986i	$-0.269940\pi$
$347$	8551.14	1.32291	0.661454	+	0.749986i	$0.269940\pi$
$348$	0	0
$349$	−5411.94	−0.830070	−0.415035	−	0.909805i	$-0.636231\pi$
$349$	−5411.94	−0.830070	−0.415035	+	0.909805i	$0.636231\pi$
$350$	0	0
$351$	−622.782	−0.0947055
$352$	0	0
$353$	11454.0	1.72701	0.863505	−	0.504340i	$-0.168264\pi$
$353$	11454.0	1.72701	0.863505	+	0.504340i	$0.168264\pi$
$354$	0	0
$355$	7239.37	1.08233
$356$	0	0
$357$	−1306.88	−0.193746
$358$	0	0
$359$	268.349	0.0394511	0.0197255	−	0.999805i	$-0.493721\pi$
$359$	268.349	0.0394511	0.0197255	+	0.999805i	$0.493721\pi$
$360$	0	0
$361$	−4021.68	−0.586336
$362$	0	0
$363$	10056.1	1.45402
$364$	0	0
$365$	−6423.21	−0.921113
$366$	0	0
$367$	−5471.90	−0.778286	−0.389143	−	0.921177i	$-0.627229\pi$
$367$	−5471.90	−0.778286	−0.389143	+	0.921177i	$0.627229\pi$
$368$	0	0
$369$	660.872	0.0932348
$370$	0	0
$371$	−3273.16	−0.458043
$372$	0	0
$373$	−6921.68	−0.960833	−0.480417	−	0.877040i	$-0.659515\pi$
$373$	−6921.68	−0.960833	−0.480417	+	0.877040i	$0.659515\pi$
$374$	0	0
$375$	4368.16	0.601522
$376$	0	0
$377$	−997.985	−0.136336
$378$	0	0
$379$	11178.3	1.51502	0.757508	−	0.652826i	$-0.226417\pi$
$379$	11178.3	1.51502	0.757508	+	0.652826i	$0.226417\pi$
$380$	0	0
$381$	4089.37	0.549882
$382$	0	0
$383$	13507.1	1.80204	0.901020	−	0.433779i	$-0.142820\pi$
$383$	13507.1	1.80204	0.901020	+	0.433779i	$0.142820\pi$
$384$	0	0
$385$	5093.64	0.674275
$386$	0	0
$387$	3395.97	0.446064
$388$	0	0
$389$	12818.4	1.67075	0.835373	−	0.549684i	$-0.185252\pi$
$389$	12818.4	1.67075	0.835373	+	0.549684i	$0.185252\pi$
$390$	0	0
$391$	−5594.66	−0.723617
$392$	0	0
$393$	−3345.59	−0.429422
$394$	0	0
$395$	−3499.82	−0.445811
$396$	0	0
$397$	7832.26	0.990151	0.495076	−	0.868850i	$-0.335140\pi$
$397$	7832.26	0.990151	0.495076	+	0.868850i	$0.335140\pi$
$398$	0	0
$399$	−1118.60	−0.140351
$400$	0	0
$401$	−7603.91	−0.946936	−0.473468	−	0.880811i	$-0.656998\pi$
$401$	−7603.91	−0.946936	−0.473468	+	0.880811i	$0.656998\pi$
$402$	0	0
$403$	−2369.65	−0.292905
$404$	0	0
$405$	−861.293	−0.105674
$406$	0	0
$407$	−20684.8	−2.51918
$408$	0	0
$409$	4190.24	0.506587	0.253293	−	0.967389i	$-0.418486\pi$
$409$	4190.24	0.506587	0.253293	+	0.967389i	$0.418486\pi$
$410$	0	0
$411$	1084.61	0.130170
$412$	0	0
$413$	3026.29	0.360567
$414$	0	0
$415$	8875.25	1.04981
$416$	0	0
$417$	−1279.63	−0.150273
$418$	0	0
$419$	−8650.38	−1.00859	−0.504294	−	0.863532i	$-0.668247\pi$
$419$	−8650.38	−1.00859	−0.504294	+	0.863532i	$0.668247\pi$
$420$	0	0
$421$	−11932.8	−1.38140	−0.690698	−	0.723143i	$-0.742697\pi$
$421$	−11932.8	−1.38140	−0.690698	+	0.723143i	$0.742697\pi$
$422$	0	0
$423$	4388.96	0.504489
$424$	0	0
$425$	742.680	0.0847653
$426$	0	0
$427$	493.251	0.0559019
$428$	0	0
$429$	4735.41	0.532932
$430$	0	0
$431$	−7617.80	−0.851361	−0.425680	−	0.904874i	$-0.639965\pi$
$431$	−7617.80	−0.851361	−0.425680	+	0.904874i	$0.639965\pi$
$432$	0	0
$433$	289.679	0.0321504	0.0160752	−	0.999871i	$-0.494883\pi$
$433$	289.679	0.0321504	0.0160752	+	0.999871i	$0.494883\pi$
$434$	0	0
$435$	−1380.19	−0.152127
$436$	0	0
$437$	−4788.64	−0.524192
$438$	0	0
$439$	320.401	0.0348335	0.0174167	−	0.999848i	$-0.494456\pi$
$439$	320.401	0.0348335	0.0174167	+	0.999848i	$0.494456\pi$
$440$	0	0
$441$	441.000	0.0476190
$442$	0	0
$443$	−4874.58	−0.522795	−0.261398	−	0.965231i	$-0.584183\pi$
$443$	−4874.58	−0.522795	−0.261398	+	0.965231i	$0.584183\pi$
$444$	0	0
$445$	10079.1	1.07370
$446$	0	0
$447$	8584.39	0.908339
$448$	0	0
$449$	−599.497	−0.0630112	−0.0315056	−	0.999504i	$-0.510030\pi$
$449$	−599.497	−0.0630112	−0.0315056	+	0.999504i	$0.510030\pi$
$450$	0	0
$451$	−5025.03	−0.524655
$452$	0	0
$453$	−2448.38	−0.253940
$454$	0	0
$455$	1716.87	0.176896
$456$	0	0
$457$	−15202.5	−1.55611	−0.778056	−	0.628195i	$-0.783794\pi$
$457$	−15202.5	−1.55611	−0.778056	+	0.628195i	$0.783794\pi$
$458$	0	0
$459$	−1680.27	−0.170868
$460$	0	0
$461$	7069.98	0.714278	0.357139	−	0.934051i	$-0.383752\pi$
$461$	7069.98	0.714278	0.357139	+	0.934051i	$0.383752\pi$
$462$	0	0
$463$	12087.1	1.21325	0.606624	−	0.794989i	$-0.292523\pi$
$463$	12087.1	1.21325	0.606624	+	0.794989i	$0.292523\pi$
$464$	0	0
$465$	−3277.17	−0.326829
$466$	0	0
$467$	−2346.52	−0.232514	−0.116257	−	0.993219i	$-0.537090\pi$
$467$	−2346.52	−0.232514	−0.116257	+	0.993219i	$0.537090\pi$
$468$	0	0
$469$	−3330.60	−0.327916
$470$	0	0
$471$	−9730.88	−0.951965
$472$	0	0
$473$	−25821.7	−2.51011
$474$	0	0
$475$	635.683	0.0614045
$476$	0	0
$477$	−4208.35	−0.403956
$478$	0	0
$479$	−6961.37	−0.664036	−0.332018	−	0.943273i	$-0.607729\pi$
$479$	−6961.37	−0.664036	−0.332018	+	0.943273i	$0.607729\pi$
$480$	0	0
$481$	−6972.02	−0.660908
$482$	0	0
$483$	1887.89	0.177851
$484$	0	0
$485$	−7031.47	−0.658314
$486$	0	0
$487$	2005.69	0.186626	0.0933128	−	0.995637i	$-0.470254\pi$
$487$	2005.69	0.186626	0.0933128	+	0.995637i	$0.470254\pi$
$488$	0	0
$489$	10091.3	0.933218
$490$	0	0
$491$	−16154.0	−1.48476	−0.742382	−	0.669977i	$-0.766304\pi$
$491$	−16154.0	−1.48476	−0.742382	+	0.669977i	$0.766304\pi$
$492$	0	0
$493$	−2692.57	−0.245978
$494$	0	0
$495$	6548.96	0.594654
$496$	0	0
$497$	−4765.77	−0.430128
$498$	0	0
$499$	−1601.46	−0.143669	−0.0718347	−	0.997417i	$-0.522885\pi$
$499$	−1601.46	−0.143669	−0.0718347	+	0.997417i	$0.522885\pi$
$500$	0	0
$501$	4057.77	0.361852
$502$	0	0
$503$	6982.59	0.618963	0.309481	−	0.950905i	$-0.399845\pi$
$503$	6982.59	0.618963	0.309481	+	0.950905i	$0.399845\pi$
$504$	0	0
$505$	−7924.89	−0.698323
$506$	0	0
$507$	−4994.88	−0.437535
$508$	0	0
$509$	−3691.65	−0.321472	−0.160736	−	0.986997i	$-0.551387\pi$
$509$	−3691.65	−0.321472	−0.160736	+	0.986997i	$0.551387\pi$
$510$	0	0
$511$	4228.48	0.366061
$512$	0	0
$513$	−1438.20	−0.123778
$514$	0	0
$515$	−9179.27	−0.785412
$516$	0	0
$517$	−33372.1	−2.83888
$518$	0	0
$519$	10473.9	0.885844
$520$	0	0
$521$	7809.75	0.656721	0.328360	−	0.944553i	$-0.393504\pi$
$521$	7809.75	0.656721	0.328360	+	0.944553i	$0.393504\pi$
$522$	0	0
$523$	10258.3	0.857677	0.428838	−	0.903381i	$-0.358923\pi$
$523$	10258.3	0.857677	0.428838	+	0.903381i	$0.358923\pi$
$524$	0	0
$525$	−250.614	−0.0208337
$526$	0	0
$527$	−6393.34	−0.528459
$528$	0	0
$529$	−4085.04	−0.335747
$530$	0	0
$531$	3890.95	0.317990
$532$	0	0
$533$	−1693.74	−0.137644
$534$	0	0
$535$	12743.1	1.02978
$536$	0	0
$537$	−7643.46	−0.614227
$538$	0	0
$539$	−3353.20	−0.267964
$540$	0	0
$541$	8734.64	0.694144	0.347072	−	0.937839i	$-0.387176\pi$
$541$	8734.64	0.694144	0.347072	+	0.937839i	$0.387176\pi$
$542$	0	0
$543$	1074.76	0.0849400
$544$	0	0
$545$	−894.596	−0.0703125
$546$	0	0
$547$	−22581.7	−1.76512	−0.882561	−	0.470197i	$-0.844183\pi$
$547$	−22581.7	−1.76512	−0.882561	+	0.470197i	$0.844183\pi$
$548$	0	0
$549$	634.180	0.0493008
$550$	0	0
$551$	−2304.65	−0.178188
$552$	0	0
$553$	2303.98	0.177170
$554$	0	0
$555$	−9642.15	−0.737453
$556$	0	0
$557$	17694.2	1.34601	0.673003	−	0.739640i	$-0.265004\pi$
$557$	17694.2	1.34601	0.673003	+	0.739640i	$0.265004\pi$
$558$	0	0
$559$	−8703.49	−0.658530
$560$	0	0
$561$	12776.2	0.961516
$562$	0	0
$563$	−3648.96	−0.273153	−0.136577	−	0.990630i	$-0.543610\pi$
$563$	−3648.96	−0.273153	−0.136577	+	0.990630i	$0.543610\pi$
$564$	0	0
$565$	−21586.7	−1.60736
$566$	0	0
$567$	567.000	0.0419961
$568$	0	0
$569$	8165.45	0.601605	0.300803	−	0.953686i	$-0.402745\pi$
$569$	8165.45	0.601605	0.300803	+	0.953686i	$0.402745\pi$
$570$	0	0
$571$	−9544.36	−0.699508	−0.349754	−	0.936842i	$-0.613735\pi$
$571$	−9544.36	−0.699508	−0.349754	+	0.936842i	$0.613735\pi$
$572$	0	0
$573$	−6419.33	−0.468013
$574$	0	0
$575$	−1072.86	−0.0778113
$576$	0	0
$577$	9915.88	0.715431	0.357715	−	0.933831i	$-0.383556\pi$
$577$	9915.88	0.715431	0.357715	+	0.933831i	$0.383556\pi$
$578$	0	0
$579$	6550.40	0.470165
$580$	0	0
$581$	−5842.69	−0.417204
$582$	0	0
$583$	31998.7	2.27316
$584$	0	0
$585$	2207.40	0.156008
$586$	0	0
$587$	12548.4	0.882328	0.441164	−	0.897426i	$-0.354566\pi$
$587$	12548.4	0.882328	0.441164	+	0.897426i	$0.354566\pi$
$588$	0	0
$589$	−5472.25	−0.382819
$590$	0	0
$591$	4608.75	0.320776
$592$	0	0
$593$	−14944.8	−1.03493	−0.517463	−	0.855706i	$-0.673123\pi$
$593$	−14944.8	−1.03493	−0.517463	+	0.855706i	$0.673123\pi$
$594$	0	0
$595$	4632.12	0.319157
$596$	0	0
$597$	10239.5	0.701965
$598$	0	0
$599$	−1446.20	−0.0986478	−0.0493239	−	0.998783i	$-0.515707\pi$
$599$	−1446.20	−0.0986478	−0.0493239	+	0.998783i	$0.515707\pi$
$600$	0	0
$601$	−8561.02	−0.581050	−0.290525	−	0.956867i	$-0.593830\pi$
$601$	−8561.02	−0.581050	−0.290525	+	0.956867i	$0.593830\pi$
$602$	0	0
$603$	−4282.20	−0.289195
$604$	0	0
$605$	−35643.1	−2.39520
$606$	0	0
$607$	−705.590	−0.0471813	−0.0235906	−	0.999722i	$-0.507510\pi$
$607$	−705.590	−0.0471813	−0.0235906	+	0.999722i	$0.507510\pi$
$608$	0	0
$609$	908.596	0.0604568
$610$	0	0
$611$	−11248.4	−0.744783
$612$	0	0
$613$	9680.02	0.637801	0.318901	−	0.947788i	$-0.396686\pi$
$613$	9680.02	0.637801	0.318901	+	0.947788i	$0.396686\pi$
$614$	0	0
$615$	−2342.41	−0.153585
$616$	0	0
$617$	−19754.3	−1.28894	−0.644471	−	0.764629i	$-0.722922\pi$
$617$	−19754.3	−1.28894	−0.644471	+	0.764629i	$0.722922\pi$
$618$	0	0
$619$	−11766.6	−0.764041	−0.382020	−	0.924154i	$-0.624772\pi$
$619$	−11766.6	−0.764041	−0.382020	+	0.924154i	$0.624772\pi$
$620$	0	0
$621$	2427.29	0.156850
$622$	0	0
$623$	−6635.23	−0.426701
$624$	0	0
$625$	−13990.8	−0.895413
$626$	0	0
$627$	10935.5	0.696527
$628$	0	0
$629$	−18810.6	−1.19241
$630$	0	0
$631$	2830.92	0.178601	0.0893005	−	0.996005i	$-0.471537\pi$
$631$	2830.92	0.178601	0.0893005	+	0.996005i	$0.471537\pi$
$632$	0	0
$633$	−6136.72	−0.385328
$634$	0	0
$635$	−14494.4	−0.905818
$636$	0	0
$637$	−1130.23	−0.0703006
$638$	0	0
$639$	−6127.41	−0.379338
$640$	0	0
$641$	−20128.9	−1.24032	−0.620159	−	0.784476i	$-0.712932\pi$
$641$	−20128.9	−1.24032	−0.620159	+	0.784476i	$0.712932\pi$
$642$	0	0
$643$	25860.4	1.58606	0.793029	−	0.609183i	$-0.208503\pi$
$643$	25860.4	1.58606	0.793029	+	0.609183i	$0.208503\pi$
$644$	0	0
$645$	−12036.7	−0.734800
$646$	0	0
$647$	−10812.4	−0.657000	−0.328500	−	0.944504i	$-0.606543\pi$
$647$	−10812.4	−0.657000	−0.328500	+	0.944504i	$0.606543\pi$
$648$	0	0
$649$	−29585.4	−1.78941
$650$	0	0
$651$	2157.40	0.129885
$652$	0	0
$653$	16162.9	0.968610	0.484305	−	0.874899i	$-0.339073\pi$
$653$	16162.9	0.968610	0.484305	+	0.874899i	$0.339073\pi$
$654$	0	0
$655$	11858.2	0.707385
$656$	0	0
$657$	5436.62	0.322835
$658$	0	0
$659$	−23949.7	−1.41570	−0.707852	−	0.706361i	$-0.750336\pi$
$659$	−23949.7	−1.41570	−0.707852	+	0.706361i	$0.750336\pi$
$660$	0	0
$661$	5942.54	0.349679	0.174840	−	0.984597i	$-0.444059\pi$
$661$	5942.54	0.349679	0.174840	+	0.984597i	$0.444059\pi$
$662$	0	0
$663$	4306.35	0.252254
$664$	0	0
$665$	3964.77	0.231199
$666$	0	0
$667$	3889.65	0.225799
$668$	0	0
$669$	−14307.0	−0.826816
$670$	0	0
$671$	−4822.08	−0.277428
$672$	0	0
$673$	19802.4	1.13422	0.567108	−	0.823643i	$-0.308062\pi$
$673$	19802.4	1.13422	0.567108	+	0.823643i	$0.308062\pi$
$674$	0	0
$675$	−322.218	−0.0183736
$676$	0	0
$677$	2528.24	0.143528	0.0717639	−	0.997422i	$-0.477137\pi$
$677$	2528.24	0.143528	0.0717639	+	0.997422i	$0.477137\pi$
$678$	0	0
$679$	4628.90	0.261621
$680$	0	0
$681$	11452.5	0.644434
$682$	0	0
$683$	−21934.9	−1.22887	−0.614434	−	0.788968i	$-0.710616\pi$
$683$	−21934.9	−1.22887	−0.614434	+	0.788968i	$0.710616\pi$
$684$	0	0
$685$	−3844.30	−0.214428
$686$	0	0
$687$	−864.867	−0.0480302
$688$	0	0
$689$	10785.5	0.596365
$690$	0	0
$691$	11360.7	0.625444	0.312722	−	0.949845i	$-0.398759\pi$
$691$	11360.7	0.625444	0.312722	+	0.949845i	$0.398759\pi$
$692$	0	0
$693$	−4311.26	−0.236322
$694$	0	0
$695$	4535.54	0.247544
$696$	0	0
$697$	−4569.73	−0.248337
$698$	0	0
$699$	27.4638	0.00148609
$700$	0	0
$701$	−2020.57	−0.108867	−0.0544335	−	0.998517i	$-0.517335\pi$
$701$	−2020.57	−0.108867	−0.0544335	+	0.998517i	$0.517335\pi$
$702$	0	0
$703$	−16100.5	−0.863789
$704$	0	0
$705$	−15556.3	−0.831042
$706$	0	0
$707$	5217.05	0.277521
$708$	0	0
$709$	−21569.3	−1.14253	−0.571265	−	0.820766i	$-0.693547\pi$
$709$	−21569.3	−1.14253	−0.571265	+	0.820766i	$0.693547\pi$
$710$	0	0
$711$	2962.26	0.156249
$712$	0	0
$713$	9235.72	0.485106
$714$	0	0
$715$	−16784.3	−0.877896
$716$	0	0
$717$	−19891.1	−1.03605
$718$	0	0
$719$	11383.4	0.590442	0.295221	−	0.955429i	$-0.404607\pi$
$719$	11383.4	0.590442	0.295221	+	0.955429i	$0.404607\pi$
$720$	0	0
$721$	6042.83	0.312131
$722$	0	0
$723$	20181.8	1.03813
$724$	0	0
$725$	−516.343	−0.0264503
$726$	0	0
$727$	22056.2	1.12520	0.562600	−	0.826730i	$-0.309801\pi$
$727$	22056.2	1.12520	0.562600	+	0.826730i	$0.309801\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	−23482.1	−1.18812
$732$	0	0
$733$	29711.0	1.49713	0.748567	−	0.663059i	$-0.230742\pi$
$733$	29711.0	1.49713	0.748567	+	0.663059i	$0.230742\pi$
$734$	0	0
$735$	−1563.09	−0.0784427
$736$	0	0
$737$	32560.3	1.62737
$738$	0	0
$739$	27088.3	1.34839	0.674195	−	0.738554i	$-0.264491\pi$
$739$	27088.3	1.34839	0.674195	+	0.738554i	$0.264491\pi$
$740$	0	0
$741$	3685.93	0.182734
$742$	0	0
$743$	−8755.28	−0.432301	−0.216151	−	0.976360i	$-0.569350\pi$
$743$	−8755.28	−0.432301	−0.216151	+	0.976360i	$0.569350\pi$
$744$	0	0
$745$	−30426.6	−1.49630
$746$	0	0
$747$	−7512.03	−0.367940
$748$	0	0
$749$	−8388.92	−0.409245
$750$	0	0
$751$	−23256.5	−1.13001	−0.565007	−	0.825086i	$-0.691126\pi$
$751$	−23256.5	−1.13001	−0.565007	+	0.825086i	$0.691126\pi$
$752$	0	0
$753$	−2633.38	−0.127445
$754$	0	0
$755$	8678.08	0.418315
$756$	0	0
$757$	14819.9	0.711544	0.355772	−	0.934573i	$-0.384218\pi$
$757$	14819.9	0.711544	0.355772	+	0.934573i	$0.384218\pi$
$758$	0	0
$759$	−18456.3	−0.882635
$760$	0	0
$761$	−19782.1	−0.942316	−0.471158	−	0.882049i	$-0.656164\pi$
$761$	−19782.1	−0.942316	−0.471158	+	0.882049i	$0.656164\pi$
$762$	0	0
$763$	588.924	0.0279430
$764$	0	0
$765$	5955.58	0.281470
$766$	0	0
$767$	−9972.06	−0.469453
$768$	0	0
$769$	29944.9	1.40421	0.702107	−	0.712071i	$-0.252243\pi$
$769$	29944.9	1.40421	0.702107	+	0.712071i	$0.252243\pi$
$770$	0	0
$771$	−18998.0	−0.887412
$772$	0	0
$773$	−30341.6	−1.41179	−0.705894	−	0.708317i	$-0.749455\pi$
$773$	−30341.6	−1.41179	−0.705894	+	0.708317i	$0.749455\pi$
$774$	0	0
$775$	−1226.02	−0.0568258
$776$	0	0
$777$	6347.54	0.293072
$778$	0	0
$779$	−3911.37	−0.179897
$780$	0	0
$781$	46590.6	2.13463
$782$	0	0
$783$	1168.20	0.0533179
$784$	0	0
$785$	34490.3	1.56817
$786$	0	0
$787$	−15196.6	−0.688309	−0.344154	−	0.938913i	$-0.611834\pi$
$787$	−15196.6	−0.688309	−0.344154	+	0.938913i	$0.611834\pi$
$788$	0	0
$789$	−10842.7	−0.489239
$790$	0	0
$791$	14210.8	0.638784
$792$	0	0
$793$	−1625.33	−0.0727835
$794$	0	0
$795$	14916.1	0.665435
$796$	0	0
$797$	4576.99	0.203419	0.101710	−	0.994814i	$-0.467569\pi$
$797$	4576.99	0.203419	0.101710	+	0.994814i	$0.467569\pi$
$798$	0	0
$799$	−30348.3	−1.34374
$800$	0	0
$801$	−8531.01	−0.376315
$802$	0	0
$803$	−41338.1	−1.81667
$804$	0	0
$805$	−6691.49	−0.292974
$806$	0	0
$807$	21344.7	0.931065
$808$	0	0
$809$	−21624.5	−0.939775	−0.469887	−	0.882726i	$-0.655705\pi$
$809$	−21624.5	−0.939775	−0.469887	+	0.882726i	$0.655705\pi$
$810$	0	0
$811$	7916.86	0.342785	0.171392	−	0.985203i	$-0.445173\pi$
$811$	7916.86	0.342785	0.171392	+	0.985203i	$0.445173\pi$
$812$	0	0
$813$	1642.14	0.0708391
$814$	0	0
$815$	−35767.7	−1.53729
$816$	0	0
$817$	−20099.0	−0.860681
$818$	0	0
$819$	−1453.16	−0.0619993
$820$	0	0
$821$	3640.19	0.154742	0.0773712	−	0.997002i	$-0.475347\pi$
$821$	3640.19	0.154742	0.0773712	+	0.997002i	$0.475347\pi$
$822$	0	0
$823$	24397.0	1.03333	0.516663	−	0.856189i	$-0.327174\pi$
$823$	24397.0	1.03333	0.516663	+	0.856189i	$0.327174\pi$
$824$	0	0
$825$	2450.03	0.103393
$826$	0	0
$827$	21546.0	0.905959	0.452979	−	0.891521i	$-0.350361\pi$
$827$	21546.0	0.905959	0.452979	+	0.891521i	$0.350361\pi$
$828$	0	0
$829$	9899.71	0.414754	0.207377	−	0.978261i	$-0.433507\pi$
$829$	9899.71	0.414754	0.207377	+	0.978261i	$0.433507\pi$
$830$	0	0
$831$	17178.4	0.717100
$832$	0	0
$833$	−3049.38	−0.126836
$834$	0	0
$835$	−14382.4	−0.596078
$836$	0	0
$837$	2773.80	0.114548
$838$	0	0
$839$	−15138.9	−0.622948	−0.311474	−	0.950255i	$-0.600823\pi$
$839$	−15138.9	−0.622948	−0.311474	+	0.950255i	$0.600823\pi$
$840$	0	0
$841$	−22517.0	−0.923244
$842$	0	0
$843$	712.135	0.0290952
$844$	0	0
$845$	17703.9	0.720750
$846$	0	0
$847$	23464.3	0.951880
$848$	0	0
$849$	9538.99	0.385603
$850$	0	0
$851$	27173.5	1.09459
$852$	0	0
$853$	35656.2	1.43124	0.715618	−	0.698492i	$-0.246145\pi$
$853$	35656.2	1.43124	0.715618	+	0.698492i	$0.246145\pi$
$854$	0	0
$855$	5097.56	0.203898
$856$	0	0
$857$	1951.71	0.0777938	0.0388969	−	0.999243i	$-0.487616\pi$
$857$	1951.71	0.0777938	0.0388969	+	0.999243i	$0.487616\pi$
$858$	0	0
$859$	41249.5	1.63843	0.819217	−	0.573483i	$-0.194408\pi$
$859$	41249.5	1.63843	0.819217	+	0.573483i	$0.194408\pi$
$860$	0	0
$861$	1542.03	0.0610365
$862$	0	0
$863$	7354.01	0.290074	0.145037	−	0.989426i	$-0.453670\pi$
$863$	7354.01	0.290074	0.145037	+	0.989426i	$0.453670\pi$
$864$	0	0
$865$	−37123.8	−1.45925
$866$	0	0
$867$	−3120.44	−0.122233
$868$	0	0
$869$	−22523.9	−0.879254
$870$	0	0
$871$	10974.8	0.426942
$872$	0	0
$873$	5951.44	0.230728
$874$	0	0
$875$	10192.4	0.393789
$876$	0	0
$877$	31367.3	1.20775	0.603875	−	0.797079i	$-0.293623\pi$
$877$	31367.3	1.20775	0.603875	+	0.797079i	$0.293623\pi$
$878$	0	0
$879$	−23272.8	−0.893027
$880$	0	0
$881$	44120.3	1.68723	0.843615	−	0.536948i	$-0.180423\pi$
$881$	44120.3	1.68723	0.843615	+	0.536948i	$0.180423\pi$
$882$	0	0
$883$	48261.2	1.83932	0.919659	−	0.392719i	$-0.128465\pi$
$883$	48261.2	1.83932	0.919659	+	0.392719i	$0.128465\pi$
$884$	0	0
$885$	−13791.1	−0.523824
$886$	0	0
$887$	−11769.9	−0.445541	−0.222770	−	0.974871i	$-0.571510\pi$
$887$	−11769.9	−0.445541	−0.222770	+	0.974871i	$0.571510\pi$
$888$	0	0
$889$	9541.87	0.359982
$890$	0	0
$891$	−5543.05	−0.208417
$892$	0	0
$893$	−25976.1	−0.973411
$894$	0	0
$895$	27091.6	1.01181
$896$	0	0
$897$	−6220.88	−0.231560
$898$	0	0
$899$	4444.92	0.164901
$900$	0	0
$901$	29099.4	1.07596
$902$	0	0
$903$	7923.93	0.292018
$904$	0	0
$905$	−3809.40	−0.139921
$906$	0	0
$907$	−11072.5	−0.405355	−0.202678	−	0.979246i	$-0.564964\pi$
$907$	−11072.5	−0.405355	−0.202678	+	0.979246i	$0.564964\pi$
$908$	0	0
$909$	6707.64	0.244751
$910$	0	0
$911$	−7014.73	−0.255114	−0.127557	−	0.991831i	$-0.540714\pi$
$911$	−7014.73	−0.255114	−0.127557	+	0.991831i	$0.540714\pi$
$912$	0	0
$913$	57118.8	2.07049
$914$	0	0
$915$	−2247.80	−0.0812131
$916$	0	0
$917$	−7806.39	−0.281123
$918$	0	0
$919$	10087.5	0.362086	0.181043	−	0.983475i	$-0.442053\pi$
$919$	10087.5	0.362086	0.181043	+	0.983475i	$0.442053\pi$
$920$	0	0
$921$	−25761.0	−0.921667
$922$	0	0
$923$	15703.9	0.560021
$924$	0	0
$925$	−3607.22	−0.128221
$926$	0	0
$927$	7769.35	0.275274
$928$	0	0
$929$	3648.41	0.128849	0.0644243	−	0.997923i	$-0.479479\pi$
$929$	3648.41	0.128849	0.0644243	+	0.997923i	$0.479479\pi$
$930$	0	0
$931$	−2610.06	−0.0918810
$932$	0	0
$933$	−11668.3	−0.409435
$934$	0	0
$935$	−45284.1	−1.58390
$936$	0	0
$937$	−38160.4	−1.33047	−0.665233	−	0.746636i	$-0.731668\pi$
$937$	−38160.4	−1.33047	−0.665233	+	0.746636i	$0.731668\pi$
$938$	0	0
$939$	25226.5	0.876716
$940$	0	0
$941$	14259.1	0.493978	0.246989	−	0.969018i	$-0.420559\pi$
$941$	14259.1	0.493978	0.246989	+	0.969018i	$0.420559\pi$
$942$	0	0
$943$	6601.36	0.227964
$944$	0	0
$945$	−2009.68	−0.0691799
$946$	0	0
$947$	25268.1	0.867058	0.433529	−	0.901140i	$-0.357268\pi$
$947$	25268.1	0.867058	0.433529	+	0.901140i	$0.357268\pi$
$948$	0	0
$949$	−13933.4	−0.476606
$950$	0	0
$951$	16955.2	0.578139
$952$	0	0
$953$	−2129.49	−0.0723831	−0.0361916	−	0.999345i	$-0.511523\pi$
$953$	−2129.49	−0.0723831	−0.0361916	+	0.999345i	$0.511523\pi$
$954$	0	0
$955$	22752.8	0.770956
$956$	0	0
$957$	−8882.54	−0.300033
$958$	0	0
$959$	2530.75	0.0852160
$960$	0	0
$961$	−19236.8	−0.645726
$962$	0	0
$963$	−10785.8	−0.360920
$964$	0	0
$965$	−23217.3	−0.774500
$966$	0	0
$967$	−56049.1	−1.86393	−0.931963	−	0.362554i	$-0.881905\pi$
$967$	−56049.1	−1.86393	−0.931963	+	0.362554i	$0.881905\pi$
$968$	0	0
$969$	9944.68	0.329690
$970$	0	0
$971$	55037.1	1.81898	0.909488	−	0.415730i	$-0.136474\pi$
$971$	55037.1	1.81898	0.909488	+	0.415730i	$0.136474\pi$
$972$	0	0
$973$	−2985.80	−0.0983766
$974$	0	0
$975$	825.809	0.0271252
$976$	0	0
$977$	41094.1	1.34567	0.672833	−	0.739794i	$-0.265077\pi$
$977$	41094.1	1.34567	0.672833	+	0.739794i	$0.265077\pi$
$978$	0	0
$979$	64866.7	2.11762
$980$	0	0
$981$	757.188	0.0246434
$982$	0	0
$983$	−60115.9	−1.95056	−0.975279	−	0.220975i	$-0.929076\pi$
$983$	−60115.9	−1.95056	−0.975279	+	0.220975i	$0.929076\pi$
$984$	0	0
$985$	−16335.3	−0.528413
$986$	0	0
$987$	10240.9	0.330265
$988$	0	0
$989$	33921.9	1.09065
$990$	0	0
$991$	−13233.5	−0.424194	−0.212097	−	0.977249i	$-0.568029\pi$
$991$	−13233.5	−0.424194	−0.212097	+	0.977249i	$0.568029\pi$
$992$	0	0
$993$	23980.0	0.766346
$994$	0	0
$995$	−36292.9	−1.15634
$996$	0	0
$997$	−24562.9	−0.780256	−0.390128	−	0.920761i	$-0.627569\pi$
$997$	−24562.9	−0.780256	−0.390128	+	0.920761i	$0.627569\pi$
$998$	0	0
$999$	8161.13	0.258465

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1344.4.a.bn.1.1		2
4.3	odd	2		1344.4.a.bf.1.1		2
8.3	odd	2		672.4.a.l.1.2	yes	2
8.5	even	2		672.4.a.g.1.2	✓	2
24.5	odd	2		2016.4.a.l.1.1		2
24.11	even	2		2016.4.a.k.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
672.4.a.g.1.2	✓	2	8.5	even	2
672.4.a.l.1.2	yes	2	8.3	odd	2
1344.4.a.bf.1.1		2	4.3	odd	2
1344.4.a.bn.1.1		2	1.1	even	1	trivial
2016.4.a.k.1.1		2	24.11	even	2
2016.4.a.l.1.1		2	24.5	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:	\( N \)	\(=\)	\( 1344 = 2^{6} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1344.a (trivial)

\(f(q)\)	\(=\)	\(q+3.00000 q^{3} -10.6332 q^{5} +7.00000 q^{7} +9.00000 q^{9} +O(q^{10})\)	\(q+3.00000 q^{3} -10.6332 q^{5} +7.00000 q^{7} +9.00000 q^{9} -68.4327 q^{11} -23.0660 q^{13} -31.8997 q^{15} -62.2322 q^{17} -53.2665 q^{19} +21.0000 q^{21} +89.8997 q^{23} -11.9340 q^{25} +27.0000 q^{27} +43.2665 q^{29} +102.734 q^{31} -205.298 q^{33} -74.4327 q^{35} +302.264 q^{37} -69.1980 q^{39} +73.4302 q^{41} +377.330 q^{43} -95.6992 q^{45} +487.662 q^{47} +49.0000 q^{49} -186.697 q^{51} -467.594 q^{53} +727.662 q^{55} -159.799 q^{57} +432.327 q^{59} +70.4645 q^{61} +63.0000 q^{63} +245.266 q^{65} -475.799 q^{67} +269.699 q^{69} -680.824 q^{71} +604.069 q^{73} -35.8020 q^{75} -479.029 q^{77} +329.140 q^{79} +81.0000 q^{81} -834.670 q^{83} +661.731 q^{85} +129.799 q^{87} -947.890 q^{89} -161.462 q^{91} +308.201 q^{93} +566.396 q^{95} +661.272 q^{97} -615.895 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 6 q^{3} - 8 q^{5} + 14 q^{7} + 18 q^{9}+O(q^{10}) \) 2 * q + 6 * q^3 - 8 * q^5 + 14 * q^7 + 18 * q^9	\( 2 q + 6 q^{3} - 8 q^{5} + 14 q^{7} + 18 q^{9} - 44 q^{11} + 60 q^{13} - 24 q^{15} + 48 q^{17} - 80 q^{19} + 42 q^{21} + 140 q^{23} - 130 q^{25} + 54 q^{27} + 60 q^{29} + 232 q^{31} - 132 q^{33} - 56 q^{35} + 180 q^{37} + 180 q^{39} - 344 q^{41} + 224 q^{43} - 72 q^{45} + 312 q^{47} + 98 q^{49} + 144 q^{51} + 20 q^{53} + 792 q^{55} - 240 q^{57} - 64 q^{59} - 204 q^{61} + 126 q^{63} + 464 q^{65} - 872 q^{67} + 420 q^{69} + 164 q^{71} + 1500 q^{73} - 390 q^{75} - 308 q^{77} + 1640 q^{79} + 162 q^{81} - 2200 q^{83} + 952 q^{85} + 180 q^{87} - 264 q^{89} + 420 q^{91} + 696 q^{93} + 496 q^{95} + 2092 q^{97} - 396 q^{99}+O(q^{100}) \) 2 * q + 6 * q^3 - 8 * q^5 + 14 * q^7 + 18 * q^9 - 44 * q^11 + 60 * q^13 - 24 * q^15 + 48 * q^17 - 80 * q^19 + 42 * q^21 + 140 * q^23 - 130 * q^25 + 54 * q^27 + 60 * q^29 + 232 * q^31 - 132 * q^33 - 56 * q^35 + 180 * q^37 + 180 * q^39 - 344 * q^41 + 224 * q^43 - 72 * q^45 + 312 * q^47 + 98 * q^49 + 144 * q^51 + 20 * q^53 + 792 * q^55 - 240 * q^57 - 64 * q^59 - 204 * q^61 + 126 * q^63 + 464 * q^65 - 872 * q^67 + 420 * q^69 + 164 * q^71 + 1500 * q^73 - 390 * q^75 - 308 * q^77 + 1640 * q^79 + 162 * q^81 - 2200 * q^83 + 952 * q^85 + 180 * q^87 - 264 * q^89 + 420 * q^91 + 696 * q^93 + 496 * q^95 + 2092 * q^97 - 396 * q^99

Introduction

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