LMFDB - Embedded newform 1638.4.a.ba.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1638,4,Mod(1,1638)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1638.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1638, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 1638 = 2 \cdot 3^{2} \cdot 7 \cdot 13 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1638.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,6,0,12,-13,0,21,24,0,-26,-17] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$96.6451285894$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.360321.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 153x - 224 $ x^3 - 153*x - 224
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 546)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.48548$ of defining polynomial
Character	$\chi$	$=$	1638.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+2.00000 q^{2} +4.00000 q^{4} -20.7180 q^{5} +7.00000 q^{7} +8.00000 q^{8} -41.4360 q^{10} +4.77607 q^{11} -13.0000 q^{13} +14.0000 q^{14} +16.0000 q^{16} -4.36916 q^{17} +120.503 q^{19} -82.8719 q^{20} +9.55215 q^{22} -97.8632 q^{23} +304.235 q^{25} -26.0000 q^{26} +28.0000 q^{28} +162.421 q^{29} -140.814 q^{31} +32.0000 q^{32} -8.73832 q^{34} -145.026 q^{35} +331.624 q^{37} +241.006 q^{38} -165.744 q^{40} +168.941 q^{41} -488.182 q^{43} +19.1043 q^{44} -195.726 q^{46} -305.040 q^{47} +49.0000 q^{49} +608.469 q^{50} -52.0000 q^{52} -200.838 q^{53} -98.9506 q^{55} +56.0000 q^{56} +324.842 q^{58} -247.331 q^{59} -535.385 q^{61} -281.628 q^{62} +64.0000 q^{64} +269.334 q^{65} -677.423 q^{67} -17.4766 q^{68} -290.052 q^{70} +183.364 q^{71} -505.997 q^{73} +663.249 q^{74} +482.011 q^{76} +33.4325 q^{77} +989.806 q^{79} -331.488 q^{80} +337.882 q^{82} -787.139 q^{83} +90.5202 q^{85} -976.364 q^{86} +38.2086 q^{88} -1034.29 q^{89} -91.0000 q^{91} -391.453 q^{92} -610.079 q^{94} -2496.57 q^{95} +324.290 q^{97} +98.0000 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 6 q^{2} + 12 q^{4} - 13 q^{5} + 21 q^{7} + 24 q^{8} - 26 q^{10} - 17 q^{11} - 39 q^{13} + 42 q^{14} + 48 q^{16} - 89 q^{17} + 89 q^{19} - 52 q^{20} - 34 q^{22} - 289 q^{23} + 86 q^{25} - 78 q^{26}+ \cdots + 294 q^{98}+O(q^{100}) $ 3 * q + 6 * q^2 + 12 * q^4 - 13 * q^5 + 21 * q^7 + 24 * q^8 - 26 * q^10 - 17 * q^11 - 39 * q^13 + 42 * q^14 + 48 * q^16 - 89 * q^17 + 89 * q^19 - 52 * q^20 - 34 * q^22 - 289 * q^23 + 86 * q^25 - 78 * q^26 + 84 * q^28 - 125 * q^29 - 208 * q^31 + 96 * q^32 - 178 * q^34 - 91 * q^35 + 213 * q^37 + 178 * q^38 - 104 * q^40 - 530 * q^41 - 75 * q^43 - 68 * q^44 - 578 * q^46 - 298 * q^47 + 147 * q^49 + 172 * q^50 - 156 * q^52 - 710 * q^53 - 87 * q^55 + 168 * q^56 - 250 * q^58 + 250 * q^59 - 511 * q^61 - 416 * q^62 + 192 * q^64 + 169 * q^65 - 522 * q^67 - 356 * q^68 - 182 * q^70 - 282 * q^71 - 701 * q^73 + 426 * q^74 + 356 * q^76 - 119 * q^77 + 1262 * q^79 - 208 * q^80 - 1060 * q^82 - 1700 * q^83 - 385 * q^85 - 150 * q^86 - 136 * q^88 - 1628 * q^89 - 273 * q^91 - 1156 * q^92 - 596 * q^94 - 2775 * q^95 + 1526 * q^97 + 294 * q^98

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 $

$2$	2.00000	0.707107
$2$	2.00000	0.707107
$3$	0	0
$3$	0	0
$4$	4.00000	0.500000
$5$	−20.7180	−1.85307	−0.926536	−	0.376205i	$-0.877229\pi$
$5$	−20.7180	−1.85307	−0.926536	+	0.376205i	$0.877229\pi$
$6$	0	0
$7$	7.00000	0.377964
$7$	7.00000	0.377964
$8$	8.00000	0.353553
$9$	0	0
$10$	−41.4360	−1.31032
$11$	4.77607	0.130913	0.0654564	−	0.997855i	$-0.479150\pi$
$11$	4.77607	0.130913	0.0654564	+	0.997855i	$0.479150\pi$
$12$	0	0
$13$	−13.0000	−0.277350
$13$	−13.0000	−0.277350
$14$	14.0000	0.267261
$15$	0	0
$16$	16.0000	0.250000
$17$	−4.36916	−0.0623339	−0.0311670	−	0.999514i	$-0.509922\pi$
$17$	−4.36916	−0.0623339	−0.0311670	+	0.999514i	$0.509922\pi$
$18$	0	0
$19$	120.503	1.45501	0.727506	−	0.686101i	$-0.240679\pi$
$19$	120.503	1.45501	0.727506	+	0.686101i	$0.240679\pi$
$20$	−82.8719	−0.926536
$21$	0	0
$22$	9.55215	0.0925693
$23$	−97.8632	−0.887213	−0.443606	−	0.896222i	$-0.646301\pi$
$23$	−97.8632	−0.887213	−0.443606	+	0.896222i	$0.646301\pi$
$24$	0	0
$25$	304.235	2.43388
$26$	−26.0000	−0.196116
$27$	0	0
$28$	28.0000	0.188982
$29$	162.421	1.04003	0.520014	−	0.854158i	$-0.325927\pi$
$29$	162.421	1.04003	0.520014	+	0.854158i	$0.325927\pi$
$30$	0	0
$31$	−140.814	−0.815836	−0.407918	−	0.913019i	$-0.633745\pi$
$31$	−140.814	−0.815836	−0.407918	+	0.913019i	$0.633745\pi$
$32$	32.0000	0.176777
$33$	0	0
$34$	−8.73832	−0.0440768
$35$	−145.026	−0.700396
$36$	0	0
$37$	331.624	1.47348	0.736740	−	0.676177i	$-0.236364\pi$
$37$	331.624	1.47348	0.736740	+	0.676177i	$0.236364\pi$
$38$	241.006	1.02885
$39$	0	0
$40$	−165.744	−0.655160
$41$	168.941	0.643515	0.321758	−	0.946822i	$-0.395726\pi$
$41$	168.941	0.643515	0.321758	+	0.946822i	$0.395726\pi$
$42$	0	0
$43$	−488.182	−1.73133	−0.865664	−	0.500626i	$-0.833103\pi$
$43$	−488.182	−1.73133	−0.865664	+	0.500626i	$0.833103\pi$
$44$	19.1043	0.0654564
$45$	0	0
$46$	−195.726	−0.627354
$47$	−305.040	−0.946694	−0.473347	−	0.880876i	$-0.656954\pi$
$47$	−305.040	−0.946694	−0.473347	+	0.880876i	$0.656954\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	608.469	1.72101
$51$	0	0
$52$	−52.0000	−0.138675
$53$	−200.838	−0.520513	−0.260256	−	0.965540i	$-0.583807\pi$
$53$	−200.838	−0.520513	−0.260256	+	0.965540i	$0.583807\pi$
$54$	0	0
$55$	−98.9506	−0.242591
$56$	56.0000	0.133631
$57$	0	0
$58$	324.842	0.735411
$59$	−247.331	−0.545758	−0.272879	−	0.962048i	$-0.587976\pi$
$59$	−247.331	−0.545758	−0.272879	+	0.962048i	$0.587976\pi$
$60$	0	0
$61$	−535.385	−1.12375	−0.561877	−	0.827220i	$-0.689921\pi$
$61$	−535.385	−1.12375	−0.561877	+	0.827220i	$0.689921\pi$
$62$	−281.628	−0.576883
$63$	0	0
$64$	64.0000	0.125000
$65$	269.334	0.513950
$66$	0	0
$67$	−677.423	−1.23523	−0.617615	−	0.786481i	$-0.711901\pi$
$67$	−677.423	−1.23523	−0.617615	+	0.786481i	$0.711901\pi$
$68$	−17.4766	−0.0311670
$69$	0	0
$70$	−290.052	−0.495254
$71$	183.364	0.306498	0.153249	−	0.988188i	$-0.451026\pi$
$71$	183.364	0.306498	0.153249	+	0.988188i	$0.451026\pi$
$72$	0	0
$73$	−505.997	−0.811267	−0.405634	−	0.914036i	$-0.632949\pi$
$73$	−505.997	−0.811267	−0.405634	+	0.914036i	$0.632949\pi$
$74$	663.249	1.04191
$75$	0	0
$76$	482.011	0.727506
$77$	33.4325	0.0494804
$78$	0	0
$79$	989.806	1.40964	0.704822	−	0.709384i	$-0.251027\pi$
$79$	989.806	1.40964	0.704822	+	0.709384i	$0.251027\pi$
$80$	−331.488	−0.463268
$81$	0	0
$82$	337.882	0.455034
$83$	−787.139	−1.04096	−0.520480	−	0.853874i	$-0.674247\pi$
$83$	−787.139	−1.04096	−0.520480	+	0.853874i	$0.674247\pi$
$84$	0	0
$85$	90.5202	0.115509
$86$	−976.364	−1.22423
$87$	0	0
$88$	38.2086	0.0462847
$89$	−1034.29	−1.23185	−0.615927	−	0.787803i	$-0.711218\pi$
$89$	−1034.29	−1.23185	−0.615927	+	0.787803i	$0.711218\pi$
$90$	0	0
$91$	−91.0000	−0.104828
$92$	−391.453	−0.443606
$93$	0	0
$94$	−610.079	−0.669413
$95$	−2496.57	−2.69624
$96$	0	0
$97$	324.290	0.339450	0.169725	−	0.985491i	$-0.445712\pi$
$97$	324.290	0.339450	0.169725	+	0.985491i	$0.445712\pi$
$98$	98.0000	0.101015
$99$	0	0
$100$	1216.94	1.21694
$101$	966.966	0.952641	0.476321	−	0.879272i	$-0.341970\pi$
$101$	966.966	0.952641	0.476321	+	0.879272i	$0.341970\pi$
$102$	0	0
$103$	66.4516	0.0635696	0.0317848	−	0.999495i	$-0.489881\pi$
$103$	66.4516	0.0635696	0.0317848	+	0.999495i	$0.489881\pi$
$104$	−104.000	−0.0980581
$105$	0	0
$106$	−401.675	−0.368058
$107$	428.002	0.386696	0.193348	−	0.981130i	$-0.438065\pi$
$107$	428.002	0.386696	0.193348	+	0.981130i	$0.438065\pi$
$108$	0	0
$109$	−588.563	−0.517194	−0.258597	−	0.965985i	$-0.583260\pi$
$109$	−588.563	−0.517194	−0.258597	+	0.965985i	$0.583260\pi$
$110$	−197.901	−0.171538
$111$	0	0
$112$	112.000	0.0944911
$113$	−329.616	−0.274404	−0.137202	−	0.990543i	$-0.543811\pi$
$113$	−329.616	−0.274404	−0.137202	+	0.990543i	$0.543811\pi$
$114$	0	0
$115$	2027.53	1.64407
$116$	649.684	0.520014
$117$	0	0
$118$	−494.661	−0.385909
$119$	−30.5841	−0.0235600
$120$	0	0
$121$	−1308.19	−0.982862
$122$	−1070.77	−0.794615
$123$	0	0
$124$	−563.255	−0.407918
$125$	−3713.38	−2.65708
$126$	0	0
$127$	−1255.49	−0.877219	−0.438609	−	0.898678i	$-0.644529\pi$
$127$	−1255.49	−0.877219	−0.438609	+	0.898678i	$0.644529\pi$
$128$	128.000	0.0883883
$129$	0	0
$130$	538.667	0.363417
$131$	−2414.90	−1.61062	−0.805308	−	0.592857i	$-0.798000\pi$
$131$	−2414.90	−1.61062	−0.805308	+	0.592857i	$0.798000\pi$
$132$	0	0
$133$	843.519	0.549943
$134$	−1354.85	−0.873440
$135$	0	0
$136$	−34.9533	−0.0220384
$137$	−832.178	−0.518962	−0.259481	−	0.965748i	$-0.583551\pi$
$137$	−832.178	−0.518962	−0.259481	+	0.965748i	$0.583551\pi$
$138$	0	0
$139$	2186.57	1.33426	0.667130	−	0.744941i	$-0.267522\pi$
$139$	2186.57	1.33426	0.667130	+	0.744941i	$0.267522\pi$
$140$	−580.103	−0.350198
$141$	0	0
$142$	366.729	0.216727
$143$	−62.0890	−0.0363087
$144$	0	0
$145$	−3365.03	−1.92725
$146$	−1011.99	−0.573653
$147$	0	0
$148$	1326.50	0.736740
$149$	1149.68	0.632118	0.316059	−	0.948740i	$-0.397640\pi$
$149$	1149.68	0.632118	0.316059	+	0.948740i	$0.397640\pi$
$150$	0	0
$151$	1959.00	1.05577	0.527885	−	0.849316i	$-0.322985\pi$
$151$	1959.00	1.05577	0.527885	+	0.849316i	$0.322985\pi$
$152$	964.022	0.514424
$153$	0	0
$154$	66.8650	0.0349879
$155$	2917.38	1.51180
$156$	0	0
$157$	−2796.82	−1.42172	−0.710861	−	0.703332i	$-0.751695\pi$
$157$	−2796.82	−1.42172	−0.710861	+	0.703332i	$0.751695\pi$
$158$	1979.61	0.996769
$159$	0	0
$160$	−662.975	−0.327580
$161$	−685.043	−0.335335
$162$	0	0
$163$	−2205.89	−1.05999	−0.529995	−	0.848000i	$-0.677806\pi$
$163$	−2205.89	−1.05999	−0.529995	+	0.848000i	$0.677806\pi$
$164$	675.763	0.321758
$165$	0	0
$166$	−1574.28	−0.736070
$167$	−1928.74	−0.893716	−0.446858	−	0.894605i	$-0.647457\pi$
$167$	−1928.74	−0.893716	−0.446858	+	0.894605i	$0.647457\pi$
$168$	0	0
$169$	169.000	0.0769231
$170$	181.040	0.0816774
$171$	0	0
$172$	−1952.73	−0.865664
$173$	−2857.91	−1.25597	−0.627985	−	0.778225i	$-0.716120\pi$
$173$	−2857.91	−1.25597	−0.627985	+	0.778225i	$0.716120\pi$
$174$	0	0
$175$	2129.64	0.919919
$176$	76.4172	0.0327282
$177$	0	0
$178$	−2068.59	−0.871053
$179$	3368.92	1.40673	0.703366	−	0.710828i	$-0.251680\pi$
$179$	3368.92	1.40673	0.703366	+	0.710828i	$0.251680\pi$
$180$	0	0
$181$	−785.583	−0.322607	−0.161304	−	0.986905i	$-0.551570\pi$
$181$	−785.583	−0.322607	−0.161304	+	0.986905i	$0.551570\pi$
$182$	−182.000	−0.0741249
$183$	0	0
$184$	−782.906	−0.313677
$185$	−6870.59	−2.73046
$186$	0	0
$187$	−20.8674	−0.00816031
$188$	−1220.16	−0.473347
$189$	0	0
$190$	−4993.15	−1.90653
$191$	−4263.49	−1.61516	−0.807580	−	0.589758i	$-0.799223\pi$
$191$	−4263.49	−1.61516	−0.807580	+	0.589758i	$0.799223\pi$
$192$	0	0
$193$	−2953.69	−1.10161	−0.550806	−	0.834633i	$-0.685680\pi$
$193$	−2953.69	−1.10161	−0.550806	+	0.834633i	$0.685680\pi$
$194$	648.580	0.240027
$195$	0	0
$196$	196.000	0.0714286
$197$	3269.53	1.18246	0.591230	−	0.806503i	$-0.298643\pi$
$197$	3269.53	1.18246	0.591230	+	0.806503i	$0.298643\pi$
$198$	0	0
$199$	3465.50	1.23449	0.617244	−	0.786772i	$-0.288249\pi$
$199$	3465.50	1.23449	0.617244	+	0.786772i	$0.288249\pi$
$200$	2433.88	0.860506
$201$	0	0
$202$	1933.93	0.673619
$203$	1136.95	0.393093
$204$	0	0
$205$	−3500.11	−1.19248
$206$	132.903	0.0449505
$207$	0	0
$208$	−208.000	−0.0693375
$209$	575.530	0.190480
$210$	0	0
$211$	−3533.52	−1.15288	−0.576440	−	0.817139i	$-0.695559\pi$
$211$	−3533.52	−1.15288	−0.576440	+	0.817139i	$0.695559\pi$
$212$	−803.350	−0.260256
$213$	0	0
$214$	856.004	0.273436
$215$	10114.1	3.20828
$216$	0	0
$217$	−985.697	−0.308357
$218$	−1177.13	−0.365711
$219$	0	0
$220$	−395.802	−0.121295
$221$	56.7991	0.0172883
$222$	0	0
$223$	−3614.13	−1.08529	−0.542645	−	0.839962i	$-0.682577\pi$
$223$	−3614.13	−1.08529	−0.542645	+	0.839962i	$0.682577\pi$
$224$	224.000	0.0668153
$225$	0	0
$226$	−659.232	−0.194033
$227$	−5575.95	−1.63035	−0.815173	−	0.579217i	$-0.803358\pi$
$227$	−5575.95	−1.63035	−0.815173	+	0.579217i	$0.803358\pi$
$228$	0	0
$229$	−2466.11	−0.711639	−0.355819	−	0.934555i	$-0.615798\pi$
$229$	−2466.11	−0.711639	−0.355819	+	0.934555i	$0.615798\pi$
$230$	4055.06	1.16253
$231$	0	0
$232$	1299.37	0.367705
$233$	−2555.70	−0.718581	−0.359290	−	0.933226i	$-0.616981\pi$
$233$	−2555.70	−0.718581	−0.359290	+	0.933226i	$0.616981\pi$
$234$	0	0
$235$	6319.80	1.75429
$236$	−989.322	−0.272879
$237$	0	0
$238$	−61.1682	−0.0166594
$239$	1321.35	0.357619	0.178809	−	0.983884i	$-0.442775\pi$
$239$	1321.35	0.357619	0.178809	+	0.983884i	$0.442775\pi$
$240$	0	0
$241$	−1708.80	−0.456737	−0.228369	−	0.973575i	$-0.573339\pi$
$241$	−1708.80	−0.456737	−0.228369	+	0.973575i	$0.573339\pi$
$242$	−2616.38	−0.694988
$243$	0	0
$244$	−2141.54	−0.561877
$245$	−1015.18	−0.264725
$246$	0	0
$247$	−1566.54	−0.403548
$248$	−1126.51	−0.288442
$249$	0	0
$250$	−7426.76	−1.87884
$251$	611.446	0.153762	0.0768808	−	0.997040i	$-0.475504\pi$
$251$	611.446	0.153762	0.0768808	+	0.997040i	$0.475504\pi$
$252$	0	0
$253$	−467.402	−0.116147
$254$	−2510.98	−0.620287
$255$	0	0
$256$	256.000	0.0625000
$257$	1552.15	0.376733	0.188366	−	0.982099i	$-0.439681\pi$
$257$	1552.15	0.376733	0.188366	+	0.982099i	$0.439681\pi$
$258$	0	0
$259$	2321.37	0.556923
$260$	1077.33	0.256975
$261$	0	0
$262$	−4829.80	−1.13888
$263$	4710.85	1.10450	0.552250	−	0.833679i	$-0.313770\pi$
$263$	4710.85	1.10450	0.552250	+	0.833679i	$0.313770\pi$
$264$	0	0
$265$	4160.95	0.964548
$266$	1687.04	0.388868
$267$	0	0
$268$	−2709.69	−0.617615
$269$	−12.4875	−0.00283040	−0.00141520	−	0.999999i	$-0.500450\pi$
$269$	−12.4875	−0.00283040	−0.00141520	+	0.999999i	$0.500450\pi$
$270$	0	0
$271$	−7813.28	−1.75138	−0.875688	−	0.482877i	$-0.839592\pi$
$271$	−7813.28	−1.75138	−0.875688	+	0.482877i	$0.839592\pi$
$272$	−69.9066	−0.0155835
$273$	0	0
$274$	−1664.36	−0.366961
$275$	1453.05	0.318626
$276$	0	0
$277$	8416.68	1.82566	0.912832	−	0.408335i	$-0.133890\pi$
$277$	8416.68	1.82566	0.912832	+	0.408335i	$0.133890\pi$
$278$	4373.14	0.943465
$279$	0	0
$280$	−1160.21	−0.247627
$281$	4392.41	0.932489	0.466244	−	0.884656i	$-0.345607\pi$
$281$	4392.41	0.932489	0.466244	+	0.884656i	$0.345607\pi$
$282$	0	0
$283$	−1833.24	−0.385070	−0.192535	−	0.981290i	$-0.561671\pi$
$283$	−1833.24	−0.385070	−0.192535	+	0.981290i	$0.561671\pi$
$284$	733.457	0.153249
$285$	0	0
$286$	−124.178	−0.0256741
$287$	1182.59	0.243226
$288$	0	0
$289$	−4893.91	−0.996114
$290$	−6730.07	−1.36277
$291$	0	0
$292$	−2023.99	−0.405634
$293$	8580.80	1.71091	0.855454	−	0.517879i	$-0.173278\pi$
$293$	8580.80	1.71091	0.855454	+	0.517879i	$0.173278\pi$
$294$	0	0
$295$	5124.19	1.01133
$296$	2653.00	0.520954
$297$	0	0
$298$	2299.36	0.446975
$299$	1272.22	0.246069
$300$	0	0
$301$	−3417.28	−0.654380
$302$	3918.00	0.746542
$303$	0	0
$304$	1928.04	0.363753
$305$	11092.1	2.08240
$306$	0	0
$307$	1026.30	0.190794	0.0953971	−	0.995439i	$-0.469588\pi$
$307$	1026.30	0.190794	0.0953971	+	0.995439i	$0.469588\pi$
$308$	133.730	0.0247402
$309$	0	0
$310$	5834.76	1.06901
$311$	4690.73	0.855263	0.427631	−	0.903953i	$-0.359348\pi$
$311$	4690.73	0.855263	0.427631	+	0.903953i	$0.359348\pi$
$312$	0	0
$313$	774.561	0.139875	0.0699374	−	0.997551i	$-0.477720\pi$
$313$	774.561	0.139875	0.0699374	+	0.997551i	$0.477720\pi$
$314$	−5593.64	−1.00531
$315$	0	0
$316$	3959.22	0.704822
$317$	9791.97	1.73493	0.867463	−	0.497501i	$-0.165749\pi$
$317$	9791.97	1.73493	0.867463	+	0.497501i	$0.165749\pi$
$318$	0	0
$319$	775.734	0.136153
$320$	−1325.95	−0.231634
$321$	0	0
$322$	−1370.09	−0.237118
$323$	−526.496	−0.0906966
$324$	0	0
$325$	−3955.05	−0.675036
$326$	−4411.78	−0.749527
$327$	0	0
$328$	1351.53	0.227517
$329$	−2135.28	−0.357817
$330$	0	0
$331$	1986.58	0.329887	0.164943	−	0.986303i	$-0.447256\pi$
$331$	1986.58	0.329887	0.164943	+	0.986303i	$0.447256\pi$
$332$	−3148.56	−0.520480
$333$	0	0
$334$	−3857.48	−0.631953
$335$	14034.8	2.28897
$336$	0	0
$337$	591.170	0.0955581	0.0477790	−	0.998858i	$-0.484786\pi$
$337$	591.170	0.0955581	0.0477790	+	0.998858i	$0.484786\pi$
$338$	338.000	0.0543928
$339$	0	0
$340$	362.081	0.0577547
$341$	−672.537	−0.106803
$342$	0	0
$343$	343.000	0.0539949
$344$	−3905.46	−0.612117
$345$	0	0
$346$	−5715.82	−0.888105
$347$	−10753.2	−1.66358	−0.831792	−	0.555087i	$-0.812685\pi$
$347$	−10753.2	−1.66358	−0.831792	+	0.555087i	$0.812685\pi$
$348$	0	0
$349$	5368.78	0.823450	0.411725	−	0.911308i	$-0.364926\pi$
$349$	5368.78	0.823450	0.411725	+	0.911308i	$0.364926\pi$
$350$	4259.29	0.650481
$351$	0	0
$352$	152.834	0.0231423
$353$	−8940.31	−1.34800	−0.674001	−	0.738731i	$-0.735426\pi$
$353$	−8940.31	−1.34800	−0.674001	+	0.738731i	$0.735426\pi$
$354$	0	0
$355$	−3798.94	−0.567963
$356$	−4137.18	−0.615927
$357$	0	0
$358$	6737.84	0.994709
$359$	8280.74	1.21738	0.608692	−	0.793407i	$-0.291695\pi$
$359$	8280.74	1.21738	0.608692	+	0.793407i	$0.291695\pi$
$360$	0	0
$361$	7661.92	1.11706
$362$	−1571.17	−0.228118
$363$	0	0
$364$	−364.000	−0.0524142
$365$	10483.2	1.50334
$366$	0	0
$367$	4595.69	0.653660	0.326830	−	0.945083i	$-0.394020\pi$
$367$	4595.69	0.653660	0.326830	+	0.945083i	$0.394020\pi$
$368$	−1565.81	−0.221803
$369$	0	0
$370$	−13741.2	−1.93073
$371$	−1405.86	−0.196735
$372$	0	0
$373$	−7309.76	−1.01470	−0.507352	−	0.861739i	$-0.669376\pi$
$373$	−7309.76	−1.01470	−0.507352	+	0.861739i	$0.669376\pi$
$374$	−41.7349	−0.00577021
$375$	0	0
$376$	−2440.32	−0.334707
$377$	−2111.47	−0.288452
$378$	0	0
$379$	4167.35	0.564809	0.282404	−	0.959295i	$-0.408868\pi$
$379$	4167.35	0.564809	0.282404	+	0.959295i	$0.408868\pi$
$380$	−9986.30	−1.34812
$381$	0	0
$382$	−8526.99	−1.14209
$383$	−5265.63	−0.702509	−0.351255	−	0.936280i	$-0.614245\pi$
$383$	−5265.63	−0.702509	−0.351255	+	0.936280i	$0.614245\pi$
$384$	0	0
$385$	−692.654	−0.0916907
$386$	−5907.37	−0.778957
$387$	0	0
$388$	1297.16	0.169725
$389$	1642.30	0.214056	0.107028	−	0.994256i	$-0.465867\pi$
$389$	1642.30	0.214056	0.107028	+	0.994256i	$0.465867\pi$
$390$	0	0
$391$	427.580	0.0553035
$392$	392.000	0.0505076
$393$	0	0
$394$	6539.06	0.836125
$395$	−20506.8	−2.61217
$396$	0	0
$397$	−12656.2	−1.60000	−0.799998	−	0.600002i	$-0.795166\pi$
$397$	−12656.2	−1.60000	−0.799998	+	0.600002i	$0.795166\pi$
$398$	6931.01	0.872915
$399$	0	0
$400$	4867.76	0.608469
$401$	−6157.80	−0.766848	−0.383424	−	0.923572i	$-0.625255\pi$
$401$	−6157.80	−0.766848	−0.383424	+	0.923572i	$0.625255\pi$
$402$	0	0
$403$	1830.58	0.226272
$404$	3867.87	0.476321
$405$	0	0
$406$	2273.89	0.277959
$407$	1583.86	0.192897
$408$	0	0
$409$	−5672.86	−0.685831	−0.342916	−	0.939366i	$-0.611414\pi$
$409$	−5672.86	−0.685831	−0.342916	+	0.939366i	$0.611414\pi$
$410$	−7000.22	−0.843211
$411$	0	0
$412$	265.806	0.0317848
$413$	−1731.31	−0.206277
$414$	0	0
$415$	16307.9	1.92898
$416$	−416.000	−0.0490290
$417$	0	0
$418$	1151.06	0.134689
$419$	−4857.85	−0.566399	−0.283200	−	0.959061i	$-0.591396\pi$
$419$	−4857.85	−0.566399	−0.283200	+	0.959061i	$0.591396\pi$
$420$	0	0
$421$	8200.60	0.949342	0.474671	−	0.880163i	$-0.342567\pi$
$421$	8200.60	0.949342	0.474671	+	0.880163i	$0.342567\pi$
$422$	−7067.05	−0.815210
$423$	0	0
$424$	−1606.70	−0.184029
$425$	−1329.25	−0.151713
$426$	0	0
$427$	−3747.70	−0.424739
$428$	1712.01	0.193348
$429$	0	0
$430$	20228.3	2.26859
$431$	2797.33	0.312628	0.156314	−	0.987707i	$-0.450039\pi$
$431$	2797.33	0.312628	0.156314	+	0.987707i	$0.450039\pi$
$432$	0	0
$433$	−7104.15	−0.788461	−0.394231	−	0.919012i	$-0.628989\pi$
$433$	−7104.15	−0.788461	−0.394231	+	0.919012i	$0.628989\pi$
$434$	−1971.39	−0.218041
$435$	0	0
$436$	−2354.25	−0.258597
$437$	−11792.8	−1.29091
$438$	0	0
$439$	4528.66	0.492349	0.246175	−	0.969226i	$-0.420826\pi$
$439$	4528.66	0.492349	0.246175	+	0.969226i	$0.420826\pi$
$440$	−791.605	−0.0857688
$441$	0	0
$442$	113.598	0.0122247
$443$	−13450.5	−1.44255	−0.721276	−	0.692648i	$-0.756444\pi$
$443$	−13450.5	−1.44255	−0.721276	+	0.692648i	$0.756444\pi$
$444$	0	0
$445$	21428.5	2.28272
$446$	−7228.25	−0.767416
$447$	0	0
$448$	448.000	0.0472456
$449$	14582.1	1.53268	0.766339	−	0.642436i	$-0.222076\pi$
$449$	14582.1	1.53268	0.766339	+	0.642436i	$0.222076\pi$
$450$	0	0
$451$	806.874	0.0842444
$452$	−1318.46	−0.137202
$453$	0	0
$454$	−11151.9	−1.15283
$455$	1885.34	0.194255
$456$	0	0
$457$	3508.45	0.359121	0.179561	−	0.983747i	$-0.442532\pi$
$457$	3508.45	0.359121	0.179561	+	0.983747i	$0.442532\pi$
$458$	−4932.22	−0.503204
$459$	0	0
$460$	8110.11	0.822035
$461$	−5916.54	−0.597746	−0.298873	−	0.954293i	$-0.596611\pi$
$461$	−5916.54	−0.597746	−0.298873	+	0.954293i	$0.596611\pi$
$462$	0	0
$463$	−2768.54	−0.277894	−0.138947	−	0.990300i	$-0.544372\pi$
$463$	−2768.54	−0.277894	−0.138947	+	0.990300i	$0.544372\pi$
$464$	2598.73	0.260007
$465$	0	0
$466$	−5111.39	−0.508113
$467$	7346.67	0.727972	0.363986	−	0.931404i	$-0.381416\pi$
$467$	7346.67	0.727972	0.363986	+	0.931404i	$0.381416\pi$
$468$	0	0
$469$	−4741.96	−0.466873
$470$	12639.6	1.24047
$471$	0	0
$472$	−1978.64	−0.192954
$473$	−2331.59	−0.226653
$474$	0	0
$475$	36661.1	3.54132
$476$	−122.336	−0.0117800
$477$	0	0
$478$	2642.69	0.252875
$479$	16588.9	1.58239	0.791197	−	0.611561i	$-0.209458\pi$
$479$	16588.9	1.58239	0.791197	+	0.611561i	$0.209458\pi$
$480$	0	0
$481$	−4311.12	−0.408670
$482$	−3417.61	−0.322962
$483$	0	0
$484$	−5232.76	−0.491431
$485$	−6718.63	−0.629026
$486$	0	0
$487$	−20761.8	−1.93184	−0.965921	−	0.258836i	$-0.916661\pi$
$487$	−20761.8	−1.93184	−0.965921	+	0.258836i	$0.916661\pi$
$488$	−4283.08	−0.397307
$489$	0	0
$490$	−2030.36	−0.187189
$491$	−6810.33	−0.625959	−0.312979	−	0.949760i	$-0.601327\pi$
$491$	−6810.33	−0.625959	−0.312979	+	0.949760i	$0.601327\pi$
$492$	0	0
$493$	−709.643	−0.0648290
$494$	−3133.07	−0.285351
$495$	0	0
$496$	−2253.02	−0.203959
$497$	1283.55	0.115845
$498$	0	0
$499$	−16577.6	−1.48721	−0.743603	−	0.668622i	$-0.766885\pi$
$499$	−16577.6	−1.48721	−0.743603	+	0.668622i	$0.766885\pi$
$500$	−14853.5	−1.32854
$501$	0	0
$502$	1222.89	0.108726
$503$	−7421.37	−0.657857	−0.328929	−	0.944355i	$-0.606688\pi$
$503$	−7421.37	−0.657857	−0.328929	+	0.944355i	$0.606688\pi$
$504$	0	0
$505$	−20033.6	−1.76531
$506$	−934.804	−0.0821287
$507$	0	0
$508$	−5021.96	−0.438609
$509$	9022.10	0.785653	0.392827	−	0.919613i	$-0.371497\pi$
$509$	9022.10	0.785653	0.392827	+	0.919613i	$0.371497\pi$
$510$	0	0
$511$	−3541.98	−0.306630
$512$	512.000	0.0441942
$513$	0	0
$514$	3104.30	0.266390
$515$	−1376.74	−0.117799
$516$	0	0
$517$	−1456.89	−0.123934
$518$	4642.74	0.393804
$519$	0	0
$520$	2154.67	0.181709
$521$	−11807.5	−0.992892	−0.496446	−	0.868068i	$-0.665362\pi$
$521$	−11807.5	−0.992892	−0.496446	+	0.868068i	$0.665362\pi$
$522$	0	0
$523$	−22098.9	−1.84764	−0.923822	−	0.382823i	$-0.874952\pi$
$523$	−22098.9	−1.84764	−0.923822	+	0.382823i	$0.874952\pi$
$524$	−9659.59	−0.805308
$525$	0	0
$526$	9421.70	0.780999
$527$	615.238	0.0508543
$528$	0	0
$529$	−2589.79	−0.212854
$530$	8321.90	0.682038
$531$	0	0
$532$	3374.08	0.274971
$533$	−2196.23	−0.178479
$534$	0	0
$535$	−8867.34	−0.716576
$536$	−5419.39	−0.436720
$537$	0	0
$538$	−24.9750	−0.00200139
$539$	234.028	0.0187018
$540$	0	0
$541$	8059.75	0.640509	0.320255	−	0.947331i	$-0.396232\pi$
$541$	8059.75	0.640509	0.320255	+	0.947331i	$0.396232\pi$
$542$	−15626.6	−1.23841
$543$	0	0
$544$	−139.813	−0.0110192
$545$	12193.8	0.958397
$546$	0	0
$547$	951.002	0.0743362	0.0371681	−	0.999309i	$-0.488166\pi$
$547$	951.002	0.0743362	0.0371681	+	0.999309i	$0.488166\pi$
$548$	−3328.71	−0.259481
$549$	0	0
$550$	2906.10	0.225302
$551$	19572.2	1.51325
$552$	0	0
$553$	6928.64	0.532795
$554$	16833.4	1.29094
$555$	0	0
$556$	8746.27	0.667130
$557$	18817.6	1.43147	0.715734	−	0.698373i	$-0.246092\pi$
$557$	18817.6	1.43147	0.715734	+	0.698373i	$0.246092\pi$
$558$	0	0
$559$	6346.37	0.480184
$560$	−2320.41	−0.175099
$561$	0	0
$562$	8784.83	0.659369
$563$	10166.3	0.761029	0.380515	−	0.924775i	$-0.375747\pi$
$563$	10166.3	0.761029	0.380515	+	0.924775i	$0.375747\pi$
$564$	0	0
$565$	6828.98	0.508491
$566$	−3666.48	−0.272285
$567$	0	0
$568$	1466.91	0.108363
$569$	−5866.13	−0.432199	−0.216099	−	0.976371i	$-0.569334\pi$
$569$	−5866.13	−0.432199	−0.216099	+	0.976371i	$0.569334\pi$
$570$	0	0
$571$	18588.0	1.36232	0.681158	−	0.732137i	$-0.261477\pi$
$571$	18588.0	1.36232	0.681158	+	0.732137i	$0.261477\pi$
$572$	−248.356	−0.0181543
$573$	0	0
$574$	2365.17	0.171987
$575$	−29773.4	−2.15937
$576$	0	0
$577$	−19543.0	−1.41003	−0.705015	−	0.709192i	$-0.749060\pi$
$577$	−19543.0	−1.41003	−0.705015	+	0.709192i	$0.749060\pi$
$578$	−9787.82	−0.704359
$579$	0	0
$580$	−13460.1	−0.963623
$581$	−5509.97	−0.393446
$582$	0	0
$583$	−959.215	−0.0681417
$584$	−4047.98	−0.286826
$585$	0	0
$586$	17161.6	1.20979
$587$	21192.6	1.49014	0.745071	−	0.666986i	$-0.232416\pi$
$587$	21192.6	1.49014	0.745071	+	0.666986i	$0.232416\pi$
$588$	0	0
$589$	−16968.5	−1.18705
$590$	10248.4	0.715117
$591$	0	0
$592$	5305.99	0.368370
$593$	25022.1	1.73277	0.866385	−	0.499376i	$-0.166437\pi$
$593$	25022.1	1.73277	0.866385	+	0.499376i	$0.166437\pi$
$594$	0	0
$595$	633.641	0.0436584
$596$	4598.72	0.316059
$597$	0	0
$598$	2544.44	0.173997
$599$	−10114.9	−0.689958	−0.344979	−	0.938610i	$-0.612114\pi$
$599$	−10114.9	−0.689958	−0.344979	+	0.938610i	$0.612114\pi$
$600$	0	0
$601$	−19577.8	−1.32877	−0.664387	−	0.747389i	$-0.731307\pi$
$601$	−19577.8	−1.32877	−0.664387	+	0.747389i	$0.731307\pi$
$602$	−6834.55	−0.462717
$603$	0	0
$604$	7836.01	0.527885
$605$	27103.0	1.82131
$606$	0	0
$607$	8913.63	0.596035	0.298017	−	0.954560i	$-0.403675\pi$
$607$	8913.63	0.596035	0.298017	+	0.954560i	$0.403675\pi$
$608$	3856.09	0.257212
$609$	0	0
$610$	22184.2	1.47248
$611$	3965.51	0.262566
$612$	0	0
$613$	−19205.1	−1.26539	−0.632696	−	0.774400i	$-0.718052\pi$
$613$	−19205.1	−1.26539	−0.632696	+	0.774400i	$0.718052\pi$
$614$	2052.59	0.134912
$615$	0	0
$616$	267.460	0.0174940
$617$	23532.7	1.53548	0.767740	−	0.640762i	$-0.221381\pi$
$617$	23532.7	1.53548	0.767740	+	0.640762i	$0.221381\pi$
$618$	0	0
$619$	3863.26	0.250852	0.125426	−	0.992103i	$-0.459970\pi$
$619$	3863.26	0.250852	0.125426	+	0.992103i	$0.459970\pi$
$620$	11669.5	0.755901
$621$	0	0
$622$	9381.46	0.604762
$623$	−7240.06	−0.465597
$624$	0	0
$625$	38904.4	2.48988
$626$	1549.12	0.0989064
$627$	0	0
$628$	−11187.3	−0.710861
$629$	−1448.92	−0.0918478
$630$	0	0
$631$	18861.6	1.18996	0.594982	−	0.803739i	$-0.297159\pi$
$631$	18861.6	1.18996	0.594982	+	0.803739i	$0.297159\pi$
$632$	7918.45	0.498384
$633$	0	0
$634$	19583.9	1.22678
$635$	26011.2	1.62555
$636$	0	0
$637$	−637.000	−0.0396214
$638$	1551.47	0.0962746
$639$	0	0
$640$	−2651.90	−0.163790
$641$	−2199.27	−0.135516	−0.0677582	−	0.997702i	$-0.521585\pi$
$641$	−2199.27	−0.135516	−0.0677582	+	0.997702i	$0.521585\pi$
$642$	0	0
$643$	1133.04	0.0694910	0.0347455	−	0.999396i	$-0.488938\pi$
$643$	1133.04	0.0694910	0.0347455	+	0.999396i	$0.488938\pi$
$644$	−2740.17	−0.167667
$645$	0	0
$646$	−1052.99	−0.0641322
$647$	17693.8	1.07514	0.537571	−	0.843218i	$-0.319342\pi$
$647$	17693.8	1.07514	0.537571	+	0.843218i	$0.319342\pi$
$648$	0	0
$649$	−1181.27	−0.0714466
$650$	−7910.10	−0.477323
$651$	0	0
$652$	−8823.55	−0.529995
$653$	−20208.7	−1.21107	−0.605533	−	0.795820i	$-0.707040\pi$
$653$	−20208.7	−1.21107	−0.605533	+	0.795820i	$0.707040\pi$
$654$	0	0
$655$	50031.8	2.98459
$656$	2703.05	0.160879
$657$	0	0
$658$	−4270.55	−0.253014
$659$	−14194.4	−0.839051	−0.419526	−	0.907744i	$-0.637804\pi$
$659$	−14194.4	−0.839051	−0.419526	+	0.907744i	$0.637804\pi$
$660$	0	0
$661$	1698.75	0.0999600	0.0499800	−	0.998750i	$-0.484084\pi$
$661$	1698.75	0.0999600	0.0499800	+	0.998750i	$0.484084\pi$
$662$	3973.17	0.233265
$663$	0	0
$664$	−6297.11	−0.368035
$665$	−17476.0	−1.01908
$666$	0	0
$667$	−15895.0	−0.922726
$668$	−7714.97	−0.446858
$669$	0	0
$670$	28069.7	1.61855
$671$	−2557.04	−0.147114
$672$	0	0
$673$	−33390.7	−1.91251	−0.956253	−	0.292542i	$-0.905499\pi$
$673$	−33390.7	−1.91251	−0.956253	+	0.292542i	$0.905499\pi$
$674$	1182.34	0.0675698
$675$	0	0
$676$	676.000	0.0384615
$677$	7417.34	0.421081	0.210540	−	0.977585i	$-0.432478\pi$
$677$	7417.34	0.421081	0.210540	+	0.977585i	$0.432478\pi$
$678$	0	0
$679$	2270.03	0.128300
$680$	724.161	0.0408387
$681$	0	0
$682$	−1345.07	−0.0755214
$683$	11589.1	0.649261	0.324631	−	0.945841i	$-0.394760\pi$
$683$	11589.1	0.649261	0.324631	+	0.945841i	$0.394760\pi$
$684$	0	0
$685$	17241.1	0.961674
$686$	686.000	0.0381802
$687$	0	0
$688$	−7810.91	−0.432832
$689$	2610.89	0.144364
$690$	0	0
$691$	29229.5	1.60918	0.804588	−	0.593833i	$-0.202386\pi$
$691$	29229.5	1.60918	0.804588	+	0.593833i	$0.202386\pi$
$692$	−11431.6	−0.627985
$693$	0	0
$694$	−21506.5	−1.17633
$695$	−45301.3	−2.47248
$696$	0	0
$697$	−738.129	−0.0401128
$698$	10737.6	0.582267
$699$	0	0
$700$	8518.57	0.459960
$701$	15437.4	0.831761	0.415880	−	0.909419i	$-0.363473\pi$
$701$	15437.4	0.831761	0.415880	+	0.909419i	$0.363473\pi$
$702$	0	0
$703$	39961.7	2.14393
$704$	305.669	0.0163641
$705$	0	0
$706$	−17880.6	−0.953181
$707$	6768.76	0.360064
$708$	0	0
$709$	5732.22	0.303636	0.151818	−	0.988408i	$-0.451487\pi$
$709$	5732.22	0.303636	0.151818	+	0.988408i	$0.451487\pi$
$710$	−7597.88	−0.401610
$711$	0	0
$712$	−8274.36	−0.435526
$713$	13780.5	0.723820
$714$	0	0
$715$	1286.36	0.0672826
$716$	13475.7	0.703366
$717$	0	0
$718$	16561.5	0.860820
$719$	−9149.31	−0.474564	−0.237282	−	0.971441i	$-0.576257\pi$
$719$	−9149.31	−0.474564	−0.237282	+	0.971441i	$0.576257\pi$
$720$	0	0
$721$	465.161	0.0240270
$722$	15323.8	0.789881
$723$	0	0
$724$	−3142.33	−0.161304
$725$	49414.1	2.53130
$726$	0	0
$727$	1378.45	0.0703217	0.0351608	−	0.999382i	$-0.488806\pi$
$727$	1378.45	0.0703217	0.0351608	+	0.999382i	$0.488806\pi$
$728$	−728.000	−0.0370625
$729$	0	0
$730$	20966.5	1.06302
$731$	2132.95	0.107920
$732$	0	0
$733$	19871.4	1.00132	0.500659	−	0.865644i	$-0.333091\pi$
$733$	19871.4	1.00132	0.500659	+	0.865644i	$0.333091\pi$
$734$	9191.38	0.462207
$735$	0	0
$736$	−3131.62	−0.156839
$737$	−3235.42	−0.161707
$738$	0	0
$739$	36445.0	1.81414	0.907070	−	0.420979i	$-0.138313\pi$
$739$	36445.0	1.81414	0.907070	+	0.420979i	$0.138313\pi$
$740$	−27482.4	−1.36523
$741$	0	0
$742$	−2811.73	−0.139113
$743$	6926.69	0.342013	0.171006	−	0.985270i	$-0.445298\pi$
$743$	6926.69	0.342013	0.171006	+	0.985270i	$0.445298\pi$
$744$	0	0
$745$	−23819.1	−1.17136
$746$	−14619.5	−0.717505
$747$	0	0
$748$	−83.4697	−0.00408015
$749$	2996.01	0.146157
$750$	0	0
$751$	−21315.0	−1.03568	−0.517841	−	0.855477i	$-0.673264\pi$
$751$	−21315.0	−1.03568	−0.517841	+	0.855477i	$0.673264\pi$
$752$	−4880.63	−0.236673
$753$	0	0
$754$	−4222.94	−0.203966
$755$	−40586.6	−1.95642
$756$	0	0
$757$	17209.0	0.826253	0.413126	−	0.910674i	$-0.364437\pi$
$757$	17209.0	0.826253	0.413126	+	0.910674i	$0.364437\pi$
$758$	8334.71	0.399380
$759$	0	0
$760$	−19972.6	−0.953266
$761$	−13206.4	−0.629083	−0.314541	−	0.949244i	$-0.601851\pi$
$761$	−13206.4	−0.629083	−0.314541	+	0.949244i	$0.601851\pi$
$762$	0	0
$763$	−4119.94	−0.195481
$764$	−17054.0	−0.807580
$765$	0	0
$766$	−10531.3	−0.496749
$767$	3215.30	0.151366
$768$	0	0
$769$	−28023.4	−1.31411	−0.657055	−	0.753843i	$-0.728198\pi$
$769$	−28023.4	−1.31411	−0.657055	+	0.753843i	$0.728198\pi$
$770$	−1385.31	−0.0648351
$771$	0	0
$772$	−11814.7	−0.550806
$773$	26004.0	1.20996	0.604980	−	0.796241i	$-0.293181\pi$
$773$	26004.0	1.20996	0.604980	+	0.796241i	$0.293181\pi$
$774$	0	0
$775$	−42840.5	−1.98564
$776$	2594.32	0.120014
$777$	0	0
$778$	3284.60	0.151361
$779$	20357.8	0.936322
$780$	0	0
$781$	875.762	0.0401245
$782$	855.160	0.0391055
$783$	0	0
$784$	784.000	0.0357143
$785$	57944.4	2.63455
$786$	0	0
$787$	13013.4	0.589427	0.294713	−	0.955586i	$-0.404776\pi$
$787$	13013.4	0.589427	0.294713	+	0.955586i	$0.404776\pi$
$788$	13078.1	0.591230
$789$	0	0
$790$	−41013.6	−1.84709
$791$	−2307.31	−0.103715
$792$	0	0
$793$	6960.01	0.311674
$794$	−25312.5	−1.13137
$795$	0	0
$796$	13862.0	0.617244
$797$	−221.082	−0.00982578	−0.00491289	−	0.999988i	$-0.501564\pi$
$797$	−221.082	−0.00982578	−0.00491289	+	0.999988i	$0.501564\pi$
$798$	0	0
$799$	1332.77	0.0590111
$800$	9735.51	0.430253
$801$	0	0
$802$	−12315.6	−0.542243
$803$	−2416.68	−0.106205
$804$	0	0
$805$	14192.7	0.621400
$806$	3661.16	0.159999
$807$	0	0
$808$	7735.73	0.336809
$809$	−30683.2	−1.33345	−0.666726	−	0.745303i	$-0.732305\pi$
$809$	−30683.2	−1.33345	−0.666726	+	0.745303i	$0.732305\pi$
$810$	0	0
$811$	−9920.45	−0.429537	−0.214768	−	0.976665i	$-0.568900\pi$
$811$	−9920.45	−0.429537	−0.214768	+	0.976665i	$0.568900\pi$
$812$	4547.78	0.196547
$813$	0	0
$814$	3167.73	0.136399
$815$	45701.6	1.96424
$816$	0	0
$817$	−58827.3	−2.51910
$818$	−11345.7	−0.484956
$819$	0	0
$820$	−14000.4	−0.596240
$821$	−14724.4	−0.625926	−0.312963	−	0.949765i	$-0.601322\pi$
$821$	−14724.4	−0.625926	−0.312963	+	0.949765i	$0.601322\pi$
$822$	0	0
$823$	−20738.3	−0.878364	−0.439182	−	0.898398i	$-0.644732\pi$
$823$	−20738.3	−0.878364	−0.439182	+	0.898398i	$0.644732\pi$
$824$	531.612	0.0224752
$825$	0	0
$826$	−3462.63	−0.145860
$827$	−43370.0	−1.82361	−0.911803	−	0.410628i	$-0.865309\pi$
$827$	−43370.0	−1.82361	−0.911803	+	0.410628i	$0.865309\pi$
$828$	0	0
$829$	−19816.5	−0.830225	−0.415113	−	0.909770i	$-0.636258\pi$
$829$	−19816.5	−0.830225	−0.415113	+	0.909770i	$0.636258\pi$
$830$	32615.9	1.36399
$831$	0	0
$832$	−832.000	−0.0346688
$833$	−214.089	−0.00890485
$834$	0	0
$835$	39959.6	1.65612
$836$	2302.12	0.0952398
$837$	0	0
$838$	−9715.69	−0.400505
$839$	−18050.1	−0.742741	−0.371370	−	0.928485i	$-0.621112\pi$
$839$	−18050.1	−0.742741	−0.371370	+	0.928485i	$0.621112\pi$
$840$	0	0
$841$	1991.54	0.0816575
$842$	16401.2	0.671286
$843$	0	0
$844$	−14134.1	−0.576440
$845$	−3501.34	−0.142544
$846$	0	0
$847$	−9157.32	−0.371487
$848$	−3213.40	−0.130128
$849$	0	0
$850$	−2658.50	−0.107277
$851$	−32453.8	−1.30729
$852$	0	0
$853$	45732.3	1.83569	0.917846	−	0.396938i	$-0.129927\pi$
$853$	45732.3	1.83569	0.917846	+	0.396938i	$0.129927\pi$
$854$	−7495.39	−0.300336
$855$	0	0
$856$	3424.02	0.136718
$857$	39858.1	1.58871	0.794356	−	0.607453i	$-0.207809\pi$
$857$	39858.1	1.58871	0.794356	+	0.607453i	$0.207809\pi$
$858$	0	0
$859$	−16129.4	−0.640660	−0.320330	−	0.947306i	$-0.603794\pi$
$859$	−16129.4	−0.640660	−0.320330	+	0.947306i	$0.603794\pi$
$860$	40456.6	1.60414
$861$	0	0
$862$	5594.66	0.221061
$863$	3507.71	0.138359	0.0691794	−	0.997604i	$-0.477962\pi$
$863$	3507.71	0.138359	0.0691794	+	0.997604i	$0.477962\pi$
$864$	0	0
$865$	59210.1	2.32740
$866$	−14208.3	−0.557526
$867$	0	0
$868$	−3942.79	−0.154178
$869$	4727.39	0.184540
$870$	0	0
$871$	8806.50	0.342591
$872$	−4708.50	−0.182856
$873$	0	0
$874$	−23585.6	−0.912808
$875$	−25993.7	−1.00428
$876$	0	0
$877$	−39422.3	−1.51790	−0.758949	−	0.651150i	$-0.774287\pi$
$877$	−39422.3	−1.51790	−0.758949	+	0.651150i	$0.774287\pi$
$878$	9057.32	0.348143
$879$	0	0
$880$	−1583.21	−0.0606477
$881$	−4291.98	−0.164132	−0.0820662	−	0.996627i	$-0.526152\pi$
$881$	−4291.98	−0.164132	−0.0820662	+	0.996627i	$0.526152\pi$
$882$	0	0
$883$	28323.0	1.07944	0.539720	−	0.841844i	$-0.318530\pi$
$883$	28323.0	1.07944	0.539720	+	0.841844i	$0.318530\pi$
$884$	227.196	0.00864416
$885$	0	0
$886$	−26900.9	−1.02004
$887$	−21374.1	−0.809100	−0.404550	−	0.914516i	$-0.632572\pi$
$887$	−21374.1	−0.809100	−0.404550	+	0.914516i	$0.632572\pi$
$888$	0	0
$889$	−8788.43	−0.331558
$890$	42857.0	1.61412
$891$	0	0
$892$	−14456.5	−0.542645
$893$	−36758.1	−1.37745
$894$	0	0
$895$	−69797.2	−2.60678
$896$	896.000	0.0334077
$897$	0	0
$898$	29164.2	1.08377
$899$	−22871.1	−0.848492
$900$	0	0
$901$	877.492	0.0324456
$902$	1613.75	0.0595698
$903$	0	0
$904$	−2636.93	−0.0970165
$905$	16275.7	0.597815
$906$	0	0
$907$	7756.25	0.283949	0.141975	−	0.989870i	$-0.454655\pi$
$907$	7756.25	0.283949	0.141975	+	0.989870i	$0.454655\pi$
$908$	−22303.8	−0.815173
$909$	0	0
$910$	3770.67	0.137359
$911$	−31045.3	−1.12906	−0.564531	−	0.825412i	$-0.690943\pi$
$911$	−31045.3	−1.12906	−0.564531	+	0.825412i	$0.690943\pi$
$912$	0	0
$913$	−3759.43	−0.136275
$914$	7016.90	0.253937
$915$	0	0
$916$	−9864.44	−0.355819
$917$	−16904.3	−0.608755
$918$	0	0
$919$	9125.50	0.327554	0.163777	−	0.986497i	$-0.447632\pi$
$919$	9125.50	0.327554	0.163777	+	0.986497i	$0.447632\pi$
$920$	16220.2	0.581266
$921$	0	0
$922$	−11833.1	−0.422670
$923$	−2383.74	−0.0850072
$924$	0	0
$925$	100892.	3.58627
$926$	−5537.08	−0.196501
$927$	0	0
$928$	5197.47	0.183853
$929$	−31370.0	−1.10787	−0.553937	−	0.832558i	$-0.686875\pi$
$929$	−31370.0	−1.10787	−0.553937	+	0.832558i	$0.686875\pi$
$930$	0	0
$931$	5904.64	0.207859
$932$	−10222.8	−0.359290
$933$	0	0
$934$	14693.3	0.514754
$935$	432.331	0.0151216
$936$	0	0
$937$	12356.6	0.430815	0.215408	−	0.976524i	$-0.430892\pi$
$937$	12356.6	0.430815	0.215408	+	0.976524i	$0.430892\pi$
$938$	−9483.93	−0.330129
$939$	0	0
$940$	25279.2	0.877146
$941$	10165.5	0.352163	0.176081	−	0.984376i	$-0.443658\pi$
$941$	10165.5	0.352163	0.176081	+	0.984376i	$0.443658\pi$
$942$	0	0
$943$	−16533.1	−0.570935
$944$	−3957.29	−0.136439
$945$	0	0
$946$	−4663.19	−0.160268
$947$	−2723.10	−0.0934412	−0.0467206	−	0.998908i	$-0.514877\pi$
$947$	−2723.10	−0.0934412	−0.0467206	+	0.998908i	$0.514877\pi$
$948$	0	0
$949$	6577.97	0.225005
$950$	73322.2	2.50409
$951$	0	0
$952$	−244.673	−0.00832972
$953$	−46944.0	−1.59566	−0.797831	−	0.602882i	$-0.794019\pi$
$953$	−46944.0	−1.59566	−0.797831	+	0.602882i	$0.794019\pi$
$954$	0	0
$955$	88331.0	2.99301
$956$	5285.39	0.178809
$957$	0	0
$958$	33177.8	1.11892
$959$	−5825.25	−0.196149
$960$	0	0
$961$	−9962.47	−0.334412
$962$	−8622.24	−0.288973
$963$	0	0
$964$	−6835.21	−0.228369
$965$	61194.4	2.04137
$966$	0	0
$967$	29887.9	0.993931	0.496965	−	0.867770i	$-0.334448\pi$
$967$	29887.9	0.993931	0.496965	+	0.867770i	$0.334448\pi$
$968$	−10465.5	−0.347494
$969$	0	0
$970$	−13437.3	−0.444788
$971$	9988.34	0.330115	0.165057	−	0.986284i	$-0.447219\pi$
$971$	9988.34	0.330115	0.165057	+	0.986284i	$0.447219\pi$
$972$	0	0
$973$	15306.0	0.504303
$974$	−41523.6	−1.36602
$975$	0	0
$976$	−8566.16	−0.280939
$977$	40337.6	1.32090	0.660448	−	0.750872i	$-0.270366\pi$
$977$	40337.6	1.32090	0.660448	+	0.750872i	$0.270366\pi$
$978$	0	0
$979$	−4939.87	−0.161265
$980$	−4060.72	−0.132362
$981$	0	0
$982$	−13620.7	−0.442620
$983$	−19688.9	−0.638840	−0.319420	−	0.947613i	$-0.603488\pi$
$983$	−19688.9	−0.638840	−0.319420	+	0.947613i	$0.603488\pi$
$984$	0	0
$985$	−67738.1	−2.19118
$986$	−1419.29	−0.0458410
$987$	0	0
$988$	−6266.14	−0.201774
$989$	47775.1	1.53606
$990$	0	0
$991$	39692.6	1.27233	0.636163	−	0.771554i	$-0.280520\pi$
$991$	39692.6	1.27233	0.636163	+	0.771554i	$0.280520\pi$
$992$	−4506.04	−0.144221
$993$	0	0
$994$	2567.10	0.0819150
$995$	−71798.3	−2.28760
$996$	0	0
$997$	−58088.8	−1.84523	−0.922613	−	0.385727i	$-0.873951\pi$
$997$	−58088.8	−1.84523	−0.922613	+	0.385727i	$0.873951\pi$
$998$	−33155.2	−1.05161
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1638.4.a.ba.1.1		3
3.2	odd	2		546.4.a.n.1.3	✓	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
546.4.a.n.1.3	✓	3	3.2	odd	2
1638.4.a.ba.1.1		3	1.1	even	1	trivial

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1638 = 2 \cdot 3^{2} \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1638.a (trivial)

\(f(q)\)	\(=\)	\(q+2.00000 q^{2} +4.00000 q^{4} -20.7180 q^{5} +7.00000 q^{7} +8.00000 q^{8} -41.4360 q^{10} +4.77607 q^{11} -13.0000 q^{13} +14.0000 q^{14} +16.0000 q^{16} -4.36916 q^{17} +120.503 q^{19} -82.8719 q^{20} +9.55215 q^{22} -97.8632 q^{23} +304.235 q^{25} -26.0000 q^{26} +28.0000 q^{28} +162.421 q^{29} -140.814 q^{31} +32.0000 q^{32} -8.73832 q^{34} -145.026 q^{35} +331.624 q^{37} +241.006 q^{38} -165.744 q^{40} +168.941 q^{41} -488.182 q^{43} +19.1043 q^{44} -195.726 q^{46} -305.040 q^{47} +49.0000 q^{49} +608.469 q^{50} -52.0000 q^{52} -200.838 q^{53} -98.9506 q^{55} +56.0000 q^{56} +324.842 q^{58} -247.331 q^{59} -535.385 q^{61} -281.628 q^{62} +64.0000 q^{64} +269.334 q^{65} -677.423 q^{67} -17.4766 q^{68} -290.052 q^{70} +183.364 q^{71} -505.997 q^{73} +663.249 q^{74} +482.011 q^{76} +33.4325 q^{77} +989.806 q^{79} -331.488 q^{80} +337.882 q^{82} -787.139 q^{83} +90.5202 q^{85} -976.364 q^{86} +38.2086 q^{88} -1034.29 q^{89} -91.0000 q^{91} -391.453 q^{92} -610.079 q^{94} -2496.57 q^{95} +324.290 q^{97} +98.0000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 6 q^{2} + 12 q^{4} - 13 q^{5} + 21 q^{7} + 24 q^{8} - 26 q^{10} - 17 q^{11} - 39 q^{13} + 42 q^{14} + 48 q^{16} - 89 q^{17} + 89 q^{19} - 52 q^{20} - 34 q^{22} - 289 q^{23} + 86 q^{25} - 78 q^{26}+ \cdots + 294 q^{98}+O(q^{100}) \) 3 * q + 6 * q^2 + 12 * q^4 - 13 * q^5 + 21 * q^7 + 24 * q^8 - 26 * q^10 - 17 * q^11 - 39 * q^13 + 42 * q^14 + 48 * q^16 - 89 * q^17 + 89 * q^19 - 52 * q^20 - 34 * q^22 - 289 * q^23 + 86 * q^25 - 78 * q^26 + 84 * q^28 - 125 * q^29 - 208 * q^31 + 96 * q^32 - 178 * q^34 - 91 * q^35 + 213 * q^37 + 178 * q^38 - 104 * q^40 - 530 * q^41 - 75 * q^43 - 68 * q^44 - 578 * q^46 - 298 * q^47 + 147 * q^49 + 172 * q^50 - 156 * q^52 - 710 * q^53 - 87 * q^55 + 168 * q^56 - 250 * q^58 + 250 * q^59 - 511 * q^61 - 416 * q^62 + 192 * q^64 + 169 * q^65 - 522 * q^67 - 356 * q^68 - 182 * q^70 - 282 * q^71 - 701 * q^73 + 426 * q^74 + 356 * q^76 - 119 * q^77 + 1262 * q^79 - 208 * q^80 - 1060 * q^82 - 1700 * q^83 - 385 * q^85 - 150 * q^86 - 136 * q^88 - 1628 * q^89 - 273 * q^91 - 1156 * q^92 - 596 * q^94 - 2775 * q^95 + 1526 * q^97 + 294 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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