LMFDB - Embedded newform 1062.6.a.m.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1062,6,Mod(1,1062)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1062, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 6, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1062.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 1062 = 2 \cdot 3^{2} \cdot 59 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	1062.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:	$170.327616641$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 17196 x^{6} - 154000 x^{5} + 98085975 x^{4} + 1816612536 x^{3} - 184506058580 x^{2} + \cdots - 7060184373200 $ x^8 - 17196x^6 - 154000x^5 + 98085975x^4 + 1816612536x^3 - 184506058580x^2 - 5340414471536x - 7060184373200
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 354)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$85.8007$ of defining polynomial
Character	$\chi$	$=$	1062.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+4.00000 q^{2} +16.0000 q^{4} -90.8007 q^{5} +19.2910 q^{7} +64.0000 q^{8} +O(q^{10})$	\(q+4.00000 q^{2} +16.0000 q^{4} -90.8007 q^{5} +19.2910 q^{7} +64.0000 q^{8} -363.203 q^{10} +68.0658 q^{11} +977.925 q^{13} +77.1641 q^{14} +256.000 q^{16} +635.634 q^{17} -2266.32 q^{19} -1452.81 q^{20} +272.263 q^{22} -4987.84 q^{23} +5119.77 q^{25} +3911.70 q^{26} +308.657 q^{28} -2939.63 q^{29} +4390.90 q^{31} +1024.00 q^{32} +2542.54 q^{34} -1751.64 q^{35} +792.231 q^{37} -9065.29 q^{38} -5811.24 q^{40} +85.1653 q^{41} +5645.02 q^{43} +1089.05 q^{44} -19951.4 q^{46} +4447.34 q^{47} -16434.9 q^{49} +20479.1 q^{50} +15646.8 q^{52} +27302.3 q^{53} -6180.42 q^{55} +1234.63 q^{56} -11758.5 q^{58} +3481.00 q^{59} -54081.5 q^{61} +17563.6 q^{62} +4096.00 q^{64} -88796.3 q^{65} +1863.86 q^{67} +10170.1 q^{68} -7006.56 q^{70} -54813.8 q^{71} +17689.4 q^{73} +3168.92 q^{74} -36261.2 q^{76} +1313.06 q^{77} +69708.2 q^{79} -23245.0 q^{80} +340.661 q^{82} -22722.8 q^{83} -57716.0 q^{85} +22580.1 q^{86} +4356.21 q^{88} +51172.2 q^{89} +18865.2 q^{91} -79805.5 q^{92} +17789.4 q^{94} +205784. q^{95} +166854. q^{97} -65739.4 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 32 q^{2} + 128 q^{4} - 40 q^{5} + 181 q^{7} + 512 q^{8}+O(q^{10}) $ 8 * q + 32 * q^2 + 128 * q^4 - 40 * q^5 + 181 * q^7 + 512 * q^8	$ 8 q + 32 q^{2} + 128 q^{4} - 40 q^{5} + 181 q^{7} + 512 q^{8} - 160 q^{10} + 349 q^{11} + 121 q^{13} + 724 q^{14} + 2048 q^{16} - 437 q^{17} + 1314 q^{19} - 640 q^{20} + 1396 q^{22} - 1224 q^{23} + 9592 q^{25} + 484 q^{26} + 2896 q^{28} - 5276 q^{29} + 18332 q^{31} + 8192 q^{32} - 1748 q^{34} - 19518 q^{35} + 30331 q^{37} + 5256 q^{38} - 2560 q^{40} - 8323 q^{41} + 30851 q^{43} + 5584 q^{44} - 4896 q^{46} + 5730 q^{47} + 32295 q^{49} + 38368 q^{50} + 1936 q^{52} + 33524 q^{53} + 23660 q^{55} + 11584 q^{56} - 21104 q^{58} + 27848 q^{59} + 2692 q^{61} + 73328 q^{62} + 32768 q^{64} + 59892 q^{65} + 56244 q^{67} - 6992 q^{68} - 78072 q^{70} + 48473 q^{71} - 30796 q^{73} + 121324 q^{74} + 21024 q^{76} - 59683 q^{77} + 135513 q^{79} - 10240 q^{80} - 33292 q^{82} + 88111 q^{83} + 114418 q^{85} + 123404 q^{86} + 22336 q^{88} + 112196 q^{89} + 377433 q^{91} - 19584 q^{92} + 22920 q^{94} - 328146 q^{95} + 551378 q^{97} + 129180 q^{98}+O(q^{100}) $ 8 * q + 32 * q^2 + 128 * q^4 - 40 * q^5 + 181 * q^7 + 512 * q^8 - 160 * q^10 + 349 * q^11 + 121 * q^13 + 724 * q^14 + 2048 * q^16 - 437 * q^17 + 1314 * q^19 - 640 * q^20 + 1396 * q^22 - 1224 * q^23 + 9592 * q^25 + 484 * q^26 + 2896 * q^28 - 5276 * q^29 + 18332 * q^31 + 8192 * q^32 - 1748 * q^34 - 19518 * q^35 + 30331 * q^37 + 5256 * q^38 - 2560 * q^40 - 8323 * q^41 + 30851 * q^43 + 5584 * q^44 - 4896 * q^46 + 5730 * q^47 + 32295 * q^49 + 38368 * q^50 + 1936 * q^52 + 33524 * q^53 + 23660 * q^55 + 11584 * q^56 - 21104 * q^58 + 27848 * q^59 + 2692 * q^61 + 73328 * q^62 + 32768 * q^64 + 59892 * q^65 + 56244 * q^67 - 6992 * q^68 - 78072 * q^70 + 48473 * q^71 - 30796 * q^73 + 121324 * q^74 + 21024 * q^76 - 59683 * q^77 + 135513 * q^79 - 10240 * q^80 - 33292 * q^82 + 88111 * q^83 + 114418 * q^85 + 123404 * q^86 + 22336 * q^88 + 112196 * q^89 + 377433 * q^91 - 19584 * q^92 + 22920 * q^94 - 328146 * q^95 + 551378 * q^97 + 129180 * q^98

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 $

$2$	4.00000	0.707107
$2$	4.00000	0.707107
$3$	0	0
$3$	0	0
$4$	16.0000	0.500000
$5$	−90.8007	−1.62429	−0.812146	−	0.583454i	$-0.801701\pi$
$5$	−90.8007	−1.62429	−0.812146	+	0.583454i	$0.801701\pi$
$6$	0	0
$7$	19.2910	0.148803	0.0744013	−	0.997228i	$-0.476295\pi$
$7$	19.2910	0.148803	0.0744013	+	0.997228i	$0.476295\pi$
$8$	64.0000	0.353553
$9$	0	0
$10$	−363.203	−1.14855
$11$	68.0658	0.169608	0.0848041	−	0.996398i	$-0.472974\pi$
$11$	68.0658	0.169608	0.0848041	+	0.996398i	$0.472974\pi$
$12$	0	0
$13$	977.925	1.60490	0.802449	−	0.596721i	$-0.203530\pi$
$13$	977.925	1.60490	0.802449	+	0.596721i	$0.203530\pi$
$14$	77.1641	0.105219
$15$	0	0
$16$	256.000	0.250000
$17$	635.634	0.533439	0.266720	−	0.963774i	$-0.414060\pi$
$17$	635.634	0.533439	0.266720	+	0.963774i	$0.414060\pi$
$18$	0	0
$19$	−2266.32	−1.44025	−0.720125	−	0.693845i	$-0.755915\pi$
$19$	−2266.32	−1.44025	−0.720125	+	0.693845i	$0.755915\pi$
$20$	−1452.81	−0.812146
$21$	0	0
$22$	272.263	0.119931
$23$	−4987.84	−1.96604	−0.983022	−	0.183490i	$-0.941261\pi$
$23$	−4987.84	−1.96604	−0.983022	+	0.183490i	$0.941261\pi$
$24$	0	0
$25$	5119.77	1.63833
$26$	3911.70	1.13483
$27$	0	0
$28$	308.657	0.0744013
$29$	−2939.63	−0.649078	−0.324539	−	0.945872i	$-0.605209\pi$
$29$	−2939.63	−0.649078	−0.324539	+	0.945872i	$0.605209\pi$
$30$	0	0
$31$	4390.90	0.820634	0.410317	−	0.911943i	$-0.365418\pi$
$31$	4390.90	0.820634	0.410317	+	0.911943i	$0.365418\pi$
$32$	1024.00	0.176777
$33$	0	0
$34$	2542.54	0.377198
$35$	−1751.64	−0.241699
$36$	0	0
$37$	792.231	0.0951365	0.0475683	−	0.998868i	$-0.484853\pi$
$37$	792.231	0.0951365	0.0475683	+	0.998868i	$0.484853\pi$
$38$	−9065.29	−1.01841
$39$	0	0
$40$	−5811.24	−0.574274
$41$	85.1653	0.00791231	0.00395615	−	0.999992i	$-0.498741\pi$
$41$	85.1653	0.00791231	0.00395615	+	0.999992i	$0.498741\pi$
$42$	0	0
$43$	5645.02	0.465580	0.232790	−	0.972527i	$-0.425215\pi$
$43$	5645.02	0.465580	0.232790	+	0.972527i	$0.425215\pi$
$44$	1089.05	0.0848041
$45$	0	0
$46$	−19951.4	−1.39020
$47$	4447.34	0.293668	0.146834	−	0.989161i	$-0.453092\pi$
$47$	4447.34	0.293668	0.146834	+	0.989161i	$0.453092\pi$
$48$	0	0
$49$	−16434.9	−0.977858
$50$	20479.1	1.15847
$51$	0	0
$52$	15646.8	0.802449
$53$	27302.3	1.33508	0.667542	−	0.744572i	$-0.267346\pi$
$53$	27302.3	1.33508	0.667542	+	0.744572i	$0.267346\pi$
$54$	0	0
$55$	−6180.42	−0.275493
$56$	1234.63	0.0526096
$57$	0	0
$58$	−11758.5	−0.458968
$59$	3481.00	0.130189
$59$	3481.00	0.130189
$60$	0	0
$61$	−54081.5	−1.86090	−0.930452	−	0.366413i	$-0.880586\pi$
$61$	−54081.5	−1.86090	−0.930452	+	0.366413i	$0.880586\pi$
$62$	17563.6	0.580276
$63$	0	0
$64$	4096.00	0.125000
$65$	−88796.3	−2.60682
$66$	0	0
$67$	1863.86	0.0507255	0.0253627	−	0.999678i	$-0.491926\pi$
$67$	1863.86	0.0507255	0.0253627	+	0.999678i	$0.491926\pi$
$68$	10170.1	0.266720
$69$	0	0
$70$	−7006.56	−0.170907
$71$	−54813.8	−1.29046	−0.645229	−	0.763989i	$-0.723238\pi$
$71$	−54813.8	−1.29046	−0.645229	+	0.763989i	$0.723238\pi$
$72$	0	0
$73$	17689.4	0.388512	0.194256	−	0.980951i	$-0.437771\pi$
$73$	17689.4	0.388512	0.194256	+	0.980951i	$0.437771\pi$
$74$	3168.92	0.0672717
$75$	0	0
$76$	−36261.2	−0.720125
$77$	1313.06	0.0252381
$78$	0	0
$79$	69708.2	1.25665	0.628327	−	0.777949i	$-0.283740\pi$
$79$	69708.2	1.25665	0.628327	+	0.777949i	$0.283740\pi$
$80$	−23245.0	−0.406073
$81$	0	0
$82$	340.661	0.00559485
$83$	−22722.8	−0.362048	−0.181024	−	0.983479i	$-0.557941\pi$
$83$	−22722.8	−0.362048	−0.181024	+	0.983479i	$0.557941\pi$
$84$	0	0
$85$	−57716.0	−0.866461
$86$	22580.1	0.329215
$87$	0	0
$88$	4356.21	0.0599656
$89$	51172.2	0.684792	0.342396	−	0.939556i	$-0.388762\pi$
$89$	51172.2	0.684792	0.342396	+	0.939556i	$0.388762\pi$
$90$	0	0
$91$	18865.2	0.238813
$92$	−79805.5	−0.983022
$93$	0	0
$94$	17789.4	0.207654
$95$	205784.	2.33939
$96$	0	0
$97$	166854.	1.80056	0.900278	−	0.435315i	$-0.143363\pi$
$97$	166854.	1.80056	0.900278	+	0.435315i	$0.143363\pi$
$98$	−65739.4	−0.691450
$99$	0	0
$100$	81916.3	0.819163
$101$	175489.	1.71177	0.855886	−	0.517164i	$-0.173012\pi$
$101$	175489.	1.71177	0.855886	+	0.517164i	$0.173012\pi$
$102$	0	0
$103$	94644.9	0.879031	0.439516	−	0.898235i	$-0.355150\pi$
$103$	94644.9	0.879031	0.439516	+	0.898235i	$0.355150\pi$
$104$	62587.2	0.567417
$105$	0	0
$106$	109209.	0.944048
$107$	−45332.0	−0.382777	−0.191388	−	0.981514i	$-0.561299\pi$
$107$	−45332.0	−0.382777	−0.191388	+	0.981514i	$0.561299\pi$
$108$	0	0
$109$	−55813.1	−0.449955	−0.224978	−	0.974364i	$-0.572231\pi$
$109$	−55813.1	−0.449955	−0.224978	+	0.974364i	$0.572231\pi$
$110$	−24721.7	−0.194803
$111$	0	0
$112$	4938.50	0.0372006
$113$	−131430.	−0.968271	−0.484136	−	0.874993i	$-0.660866\pi$
$113$	−131430.	−0.968271	−0.484136	+	0.874993i	$0.660866\pi$
$114$	0	0
$115$	452900.	3.19343
$116$	−47034.0	−0.324539
$117$	0	0
$118$	13924.0	0.0920575
$119$	12262.0	0.0793771
$120$	0	0
$121$	−156418.	−0.971233
$122$	−216326.	−1.31586
$123$	0	0
$124$	70254.4	0.410317
$125$	−181126.	−1.03683
$126$	0	0
$127$	230101.	1.26593	0.632965	−	0.774180i	$-0.281838\pi$
$127$	230101.	1.26593	0.632965	+	0.774180i	$0.281838\pi$
$128$	16384.0	0.0883883
$129$	0	0
$130$	−355185.	−1.84330
$131$	274638.	1.39824	0.699122	−	0.715002i	$-0.253574\pi$
$131$	274638.	1.39824	0.699122	+	0.715002i	$0.253574\pi$
$132$	0	0
$133$	−43719.7	−0.214313
$134$	7455.44	0.0358683
$135$	0	0
$136$	40680.6	0.188599
$137$	−260740.	−1.18688	−0.593439	−	0.804879i	$-0.702230\pi$
$137$	−260740.	−1.18688	−0.593439	+	0.804879i	$0.702230\pi$
$138$	0	0
$139$	426230.	1.87114	0.935572	−	0.353136i	$-0.114885\pi$
$139$	426230.	1.87114	0.935572	+	0.353136i	$0.114885\pi$
$140$	−28026.2	−0.120849
$141$	0	0
$142$	−219255.	−0.912492
$143$	66563.2	0.272204
$144$	0	0
$145$	266920.	1.05429
$146$	70757.4	0.274720
$147$	0	0
$148$	12675.7	0.0475683
$149$	260830.	0.962479	0.481240	−	0.876589i	$-0.340187\pi$
$149$	260830.	0.962479	0.481240	+	0.876589i	$0.340187\pi$
$150$	0	0
$151$	306281.	1.09314	0.546572	−	0.837412i	$-0.315933\pi$
$151$	306281.	1.09314	0.546572	+	0.837412i	$0.315933\pi$
$152$	−145045.	−0.509205
$153$	0	0
$154$	5252.24	0.0178461
$155$	−398697.	−1.33295
$156$	0	0
$157$	−251309.	−0.813691	−0.406846	−	0.913497i	$-0.633371\pi$
$157$	−251309.	−0.813691	−0.406846	+	0.913497i	$0.633371\pi$
$158$	278833.	0.888589
$159$	0	0
$160$	−92979.9	−0.287137
$161$	−96220.6	−0.292552
$162$	0	0
$163$	141358.	0.416726	0.208363	−	0.978052i	$-0.433186\pi$
$163$	141358.	0.416726	0.208363	+	0.978052i	$0.433186\pi$
$164$	1362.65	0.00395615
$165$	0	0
$166$	−90891.1	−0.256007
$167$	−40603.1	−0.112659	−0.0563297	−	0.998412i	$-0.517940\pi$
$167$	−40603.1	−0.112659	−0.0563297	+	0.998412i	$0.517940\pi$
$168$	0	0
$169$	585045.	1.57569
$170$	−230864.	−0.612681
$171$	0	0
$172$	90320.3	0.232790
$173$	109803.	0.278933	0.139466	−	0.990227i	$-0.455461\pi$
$173$	109803.	0.278933	0.139466	+	0.990227i	$0.455461\pi$
$174$	0	0
$175$	98765.6	0.243787
$176$	17424.8	0.0424021
$177$	0	0
$178$	204689.	0.484221
$179$	298219.	0.695669	0.347834	−	0.937556i	$-0.386917\pi$
$179$	298219.	0.695669	0.347834	+	0.937556i	$0.386917\pi$
$180$	0	0
$181$	292093.	0.662713	0.331356	−	0.943506i	$-0.392494\pi$
$181$	292093.	0.662713	0.331356	+	0.943506i	$0.392494\pi$
$182$	75460.7	0.168866
$183$	0	0
$184$	−319222.	−0.695101
$185$	−71935.1	−0.154530
$186$	0	0
$187$	43264.9	0.0904757
$188$	71157.5	0.146834
$189$	0	0
$190$	823135.	1.65420
$191$	433269.	0.859359	0.429679	−	0.902982i	$-0.358627\pi$
$191$	433269.	0.859359	0.429679	+	0.902982i	$0.358627\pi$
$192$	0	0
$193$	96493.8	0.186469	0.0932344	−	0.995644i	$-0.470279\pi$
$193$	96493.8	0.186469	0.0932344	+	0.995644i	$0.470279\pi$
$194$	667415.	1.27319
$195$	0	0
$196$	−262958.	−0.488929
$197$	842541.	1.54677	0.773385	−	0.633937i	$-0.218562\pi$
$197$	842541.	1.54677	0.773385	+	0.633937i	$0.218562\pi$
$198$	0	0
$199$	860765.	1.54082	0.770410	−	0.637549i	$-0.220051\pi$
$199$	860765.	1.54082	0.770410	+	0.637549i	$0.220051\pi$
$200$	327665.	0.579236
$201$	0	0
$202$	701955.	1.21041
$203$	−56708.4	−0.0965845
$204$	0	0
$205$	−7733.07	−0.0128519
$206$	378580.	0.621569
$207$	0	0
$208$	250349.	0.401224
$209$	−154259.	−0.244278
$210$	0	0
$211$	−233993.	−0.361824	−0.180912	−	0.983499i	$-0.557905\pi$
$211$	−233993.	−0.361824	−0.180912	+	0.983499i	$0.557905\pi$
$212$	436836.	0.667542
$213$	0	0
$214$	−181328.	−0.270664
$215$	−512571.	−0.756237
$216$	0	0
$217$	84705.0	0.122112
$218$	−223252.	−0.318167
$219$	0	0
$220$	−98886.7	−0.137747
$221$	621603.	0.856115
$222$	0	0
$223$	33977.9	0.0457545	0.0228773	−	0.999738i	$-0.492717\pi$
$223$	33977.9	0.0457545	0.0228773	+	0.999738i	$0.492717\pi$
$224$	19754.0	0.0263048
$225$	0	0
$226$	−525718.	−0.684671
$227$	−28003.6	−0.0360703	−0.0180351	−	0.999837i	$-0.505741\pi$
$227$	−28003.6	−0.0360703	−0.0180351	+	0.999837i	$0.505741\pi$
$228$	0	0
$229$	−1.38721e6	−1.74805	−0.874026	−	0.485879i	$-0.838500\pi$
$229$	−1.38721e6	−1.74805	−0.874026	+	0.485879i	$0.838500\pi$
$230$	1.81160e6	2.25810
$231$	0	0
$232$	−188136.	−0.229484
$233$	49808.4	0.0601052	0.0300526	−	0.999548i	$-0.490433\pi$
$233$	49808.4	0.0601052	0.0300526	+	0.999548i	$0.490433\pi$
$234$	0	0
$235$	−403822.	−0.477002
$236$	55696.0	0.0650945
$237$	0	0
$238$	49048.2	0.0561281
$239$	266589.	0.301889	0.150945	−	0.988542i	$-0.451768\pi$
$239$	266589.	0.301889	0.150945	+	0.988542i	$0.451768\pi$
$240$	0	0
$241$	1.05040e6	1.16496	0.582482	−	0.812844i	$-0.302082\pi$
$241$	1.05040e6	1.16496	0.582482	+	0.812844i	$0.302082\pi$
$242$	−625672.	−0.686765
$243$	0	0
$244$	−865304.	−0.930452
$245$	1.49230e6	1.58833
$246$	0	0
$247$	−2.21629e6	−2.31145
$248$	281018.	0.290138
$249$	0	0
$250$	−724505.	−0.733148
$251$	1.18382e6	1.18605	0.593025	−	0.805184i	$-0.297934\pi$
$251$	1.18382e6	1.18605	0.593025	+	0.805184i	$0.297934\pi$
$252$	0	0
$253$	−339501.	−0.333457
$254$	920405.	0.895148
$255$	0	0
$256$	65536.0	0.0625000
$257$	−1.28679e6	−1.21527	−0.607636	−	0.794215i	$-0.707882\pi$
$257$	−1.28679e6	−1.21527	−0.607636	+	0.794215i	$0.707882\pi$
$258$	0	0
$259$	15282.9	0.0141566
$260$	−1.42074e6	−1.30341
$261$	0	0
$262$	1.09855e6	0.988708
$263$	959903.	0.855733	0.427866	−	0.903842i	$-0.359265\pi$
$263$	959903.	0.855733	0.427866	+	0.903842i	$0.359265\pi$
$264$	0	0
$265$	−2.47906e6	−2.16857
$266$	−174879.	−0.151542
$267$	0	0
$268$	29821.7	0.0253627
$269$	−923814.	−0.778402	−0.389201	−	0.921153i	$-0.627249\pi$
$269$	−923814.	−0.778402	−0.389201	+	0.921153i	$0.627249\pi$
$270$	0	0
$271$	−790352.	−0.653728	−0.326864	−	0.945071i	$-0.605992\pi$
$271$	−790352.	−0.653728	−0.326864	+	0.945071i	$0.605992\pi$
$272$	162722.	0.133360
$273$	0	0
$274$	−1.04296e6	−0.839250
$275$	348481.	0.277874
$276$	0	0
$277$	−106488.	−0.0833875	−0.0416937	−	0.999130i	$-0.513275\pi$
$277$	−106488.	−0.0833875	−0.0416937	+	0.999130i	$0.513275\pi$
$278$	1.70492e6	1.32310
$279$	0	0
$280$	−112105.	−0.0854534
$281$	1.78226e6	1.34649	0.673247	−	0.739418i	$-0.264899\pi$
$281$	1.78226e6	1.34649	0.673247	+	0.739418i	$0.264899\pi$
$282$	0	0
$283$	1.85517e6	1.37695	0.688476	−	0.725259i	$-0.258280\pi$
$283$	1.85517e6	1.37695	0.688476	+	0.725259i	$0.258280\pi$
$284$	−877021.	−0.645229
$285$	0	0
$286$	266253.	0.192477
$287$	1642.93	0.00117737
$288$	0	0
$289$	−1.01583e6	−0.715443
$290$	1.06768e6	0.745497
$291$	0	0
$292$	283030.	0.194256
$293$	−1.24918e6	−0.850074	−0.425037	−	0.905176i	$-0.639739\pi$
$293$	−1.24918e6	−0.850074	−0.425037	+	0.905176i	$0.639739\pi$
$294$	0	0
$295$	−316077.	−0.211465
$296$	50702.8	0.0336358
$297$	0	0
$298$	1.04332e6	0.680576
$299$	−4.87774e6	−3.15530
$300$	0	0
$301$	108898.	0.0692794
$302$	1.22512e6	0.772969
$303$	0	0
$304$	−580179.	−0.360062
$305$	4.91064e6	3.02265
$306$	0	0
$307$	−2.08103e6	−1.26018	−0.630091	−	0.776521i	$-0.716982\pi$
$307$	−2.08103e6	−1.26018	−0.630091	+	0.776521i	$0.716982\pi$
$308$	21008.9	0.0126191
$309$	0	0
$310$	−1.59479e6	−0.942537
$311$	−2.44568e6	−1.43383	−0.716916	−	0.697159i	$-0.754447\pi$
$311$	−2.44568e6	−1.43383	−0.716916	+	0.697159i	$0.754447\pi$
$312$	0	0
$313$	3.14454e6	1.81425	0.907124	−	0.420864i	$-0.138273\pi$
$313$	3.14454e6	1.81425	0.907124	+	0.420864i	$0.138273\pi$
$314$	−1.00524e6	−0.575367
$315$	0	0
$316$	1.11533e6	0.628327
$317$	1.47321e6	0.823409	0.411705	−	0.911317i	$-0.364934\pi$
$317$	1.47321e6	0.823409	0.411705	+	0.911317i	$0.364934\pi$
$318$	0	0
$319$	−200088.	−0.110089
$320$	−371920.	−0.203037
$321$	0	0
$322$	−384882.	−0.206866
$323$	−1.44055e6	−0.768286
$324$	0	0
$325$	5.00675e6	2.62934
$326$	565431.	0.294670
$327$	0	0
$328$	5450.58	0.00279742
$329$	85793.8	0.0436985
$330$	0	0
$331$	−2.61555e6	−1.31218	−0.656089	−	0.754684i	$-0.727790\pi$
$331$	−2.61555e6	−1.31218	−0.656089	+	0.754684i	$0.727790\pi$
$332$	−363565.	−0.181024
$333$	0	0
$334$	−162412.	−0.0796623
$335$	−169240.	−0.0823930
$336$	0	0
$337$	−1.31382e6	−0.630174	−0.315087	−	0.949063i	$-0.602034\pi$
$337$	−1.31382e6	−0.630174	−0.315087	+	0.949063i	$0.602034\pi$
$338$	2.34018e6	1.11418
$339$	0	0
$340$	−923456.	−0.433231
$341$	298870.	0.139186
$342$	0	0
$343$	−641270.	−0.294310
$344$	361281.	0.164607
$345$	0	0
$346$	439212.	0.197235
$347$	2.15457e6	0.960587	0.480294	−	0.877108i	$-0.340530\pi$
$347$	2.15457e6	0.960587	0.480294	+	0.877108i	$0.340530\pi$
$348$	0	0
$349$	−122315.	−0.0537545	−0.0268773	−	0.999639i	$-0.508556\pi$
$349$	−122315.	−0.0537545	−0.0268773	+	0.999639i	$0.508556\pi$
$350$	395062.	0.172383
$351$	0	0
$352$	69699.3	0.0299828
$353$	1.52821e6	0.652750	0.326375	−	0.945240i	$-0.394173\pi$
$353$	1.52821e6	0.652750	0.326375	+	0.945240i	$0.394173\pi$
$354$	0	0
$355$	4.97713e6	2.09608
$356$	818755.	0.342396
$357$	0	0
$358$	1.19288e6	0.491912
$359$	−571327.	−0.233964	−0.116982	−	0.993134i	$-0.537322\pi$
$359$	−571327.	−0.233964	−0.116982	+	0.993134i	$0.537322\pi$
$360$	0	0
$361$	2.66012e6	1.07432
$362$	1.16837e6	0.468609
$363$	0	0
$364$	301843.	0.119406
$365$	−1.60621e6	−0.631058
$366$	0	0
$367$	−4.74418e6	−1.83864	−0.919319	−	0.393514i	$-0.871259\pi$
$367$	−4.74418e6	−1.83864	−0.919319	+	0.393514i	$0.871259\pi$
$368$	−1.27689e6	−0.491511
$369$	0	0
$370$	−287740.	−0.109269
$371$	526689.	0.198664
$372$	0	0
$373$	2.62099e6	0.975424	0.487712	−	0.873005i	$-0.337832\pi$
$373$	2.62099e6	0.975424	0.487712	+	0.873005i	$0.337832\pi$
$374$	173060.	0.0639760
$375$	0	0
$376$	284630.	0.103827
$377$	−2.87473e6	−1.04170
$378$	0	0
$379$	40496.3	0.0144816	0.00724081	−	0.999974i	$-0.497695\pi$
$379$	40496.3	0.0144816	0.00724081	+	0.999974i	$0.497695\pi$
$380$	3.29254e6	1.16969
$381$	0	0
$382$	1.73308e6	0.607658
$383$	1.86409e6	0.649335	0.324668	−	0.945828i	$-0.394748\pi$
$383$	1.86409e6	0.649335	0.324668	+	0.945828i	$0.394748\pi$
$384$	0	0
$385$	−119227.	−0.0409941
$386$	385975.	0.131853
$387$	0	0
$388$	2.66966e6	0.900278
$389$	−3.75948e6	−1.25966	−0.629830	−	0.776733i	$-0.716876\pi$
$389$	−3.75948e6	−1.25966	−0.629830	+	0.776733i	$0.716876\pi$
$390$	0	0
$391$	−3.17044e6	−1.04876
$392$	−1.05183e6	−0.345725
$393$	0	0
$394$	3.37016e6	1.09373
$395$	−6.32955e6	−2.04117
$396$	0	0
$397$	−4.67416e6	−1.48843	−0.744214	−	0.667942i	$-0.767175\pi$
$397$	−4.67416e6	−1.48843	−0.744214	+	0.667942i	$0.767175\pi$
$398$	3.44306e6	1.08952
$399$	0	0
$400$	1.31066e6	0.409581
$401$	2.81671e6	0.874745	0.437372	−	0.899280i	$-0.355909\pi$
$401$	2.81671e6	0.874745	0.437372	+	0.899280i	$0.355909\pi$
$402$	0	0
$403$	4.29397e6	1.31703
$404$	2.80782e6	0.855886
$405$	0	0
$406$	−226834.	−0.0682955
$407$	53923.8	0.0161359
$408$	0	0
$409$	3.61326e6	1.06805	0.534024	−	0.845469i	$-0.320679\pi$
$409$	3.61326e6	1.06805	0.534024	+	0.845469i	$0.320679\pi$
$410$	−30932.3	−0.00908767
$411$	0	0
$412$	1.51432e6	0.439516
$413$	67152.1	0.0193724
$414$	0	0
$415$	2.06324e6	0.588072
$416$	1.00140e6	0.283708
$417$	0	0
$418$	−617036.	−0.172731
$419$	2.23385e6	0.621610	0.310805	−	0.950474i	$-0.399401\pi$
$419$	2.23385e6	0.621610	0.310805	+	0.950474i	$0.399401\pi$
$420$	0	0
$421$	6.08814e6	1.67409	0.837047	−	0.547132i	$-0.184280\pi$
$421$	6.08814e6	1.67409	0.837047	+	0.547132i	$0.184280\pi$
$422$	−935973.	−0.255848
$423$	0	0
$424$	1.74734e6	0.472024
$425$	3.25430e6	0.873947
$426$	0	0
$427$	−1.04329e6	−0.276907
$428$	−725312.	−0.191388
$429$	0	0
$430$	−2.05029e6	−0.534741
$431$	6.26855e6	1.62545	0.812726	−	0.582646i	$-0.197982\pi$
$431$	6.26855e6	1.62545	0.812726	+	0.582646i	$0.197982\pi$
$432$	0	0
$433$	−1.47296e6	−0.377546	−0.188773	−	0.982021i	$-0.560451\pi$
$433$	−1.47296e6	−0.377546	−0.188773	+	0.982021i	$0.560451\pi$
$434$	338820.	0.0863465
$435$	0	0
$436$	−893009.	−0.224978
$437$	1.13041e7	2.83159
$438$	0	0
$439$	−2.88487e6	−0.714439	−0.357219	−	0.934021i	$-0.616275\pi$
$439$	−2.88487e6	−0.714439	−0.357219	+	0.934021i	$0.616275\pi$
$440$	−395547.	−0.0974016
$441$	0	0
$442$	2.48641e6	0.605365
$443$	−628814.	−0.152234	−0.0761172	−	0.997099i	$-0.524252\pi$
$443$	−628814.	−0.152234	−0.0761172	+	0.997099i	$0.524252\pi$
$444$	0	0
$445$	−4.64647e6	−1.11230
$446$	135911.	0.0323533
$447$	0	0
$448$	79016.1	0.0186003
$449$	6.14446e6	1.43836	0.719180	−	0.694823i	$-0.244518\pi$
$449$	6.14446e6	1.43836	0.719180	+	0.694823i	$0.244518\pi$
$450$	0	0
$451$	5796.84	0.00134199
$452$	−2.10287e6	−0.484136
$453$	0	0
$454$	−112014.	−0.0255055
$455$	−1.71297e6	−0.387902
$456$	0	0
$457$	20719.4	0.00464072	0.00232036	−	0.999997i	$-0.499261\pi$
$457$	20719.4	0.00464072	0.00232036	+	0.999997i	$0.499261\pi$
$458$	−5.54885e6	−1.23606
$459$	0	0
$460$	7.24639e6	1.59671
$461$	−4.21856e6	−0.924510	−0.462255	−	0.886747i	$-0.652960\pi$
$461$	−4.21856e6	−0.924510	−0.462255	+	0.886747i	$0.652960\pi$
$462$	0	0
$463$	1.38091e6	0.299373	0.149687	−	0.988733i	$-0.452174\pi$
$463$	1.38091e6	0.299373	0.149687	+	0.988733i	$0.452174\pi$
$464$	−752544.	−0.162270
$465$	0	0
$466$	199233.	0.0425008
$467$	3.72868e6	0.791157	0.395579	−	0.918432i	$-0.370544\pi$
$467$	3.72868e6	0.791157	0.395579	+	0.918432i	$0.370544\pi$
$468$	0	0
$469$	35955.8	0.00754808
$470$	−1.61529e6	−0.337291
$471$	0	0
$472$	222784.	0.0460287
$473$	384232.	0.0789662
$474$	0	0
$475$	−1.16030e7	−2.35960
$476$	196193.	0.0396886
$477$	0	0
$478$	1.06636e6	0.213468
$479$	894333.	0.178099	0.0890493	−	0.996027i	$-0.471617\pi$
$479$	894333.	0.178099	0.0890493	+	0.996027i	$0.471617\pi$
$480$	0	0
$481$	774742.	0.152684
$482$	4.20160e6	0.823754
$483$	0	0
$484$	−2.50269e6	−0.485617
$485$	−1.51504e7	−2.92463
$486$	0	0
$487$	7.04968e6	1.34694	0.673468	−	0.739216i	$-0.264804\pi$
$487$	7.04968e6	1.34694	0.673468	+	0.739216i	$0.264804\pi$
$488$	−3.46122e6	−0.657929
$489$	0	0
$490$	5.96919e6	1.12312
$491$	1.95206e6	0.365418	0.182709	−	0.983167i	$-0.441513\pi$
$491$	1.95206e6	0.365418	0.182709	+	0.983167i	$0.441513\pi$
$492$	0	0
$493$	−1.86853e6	−0.346244
$494$	−8.86518e6	−1.63444
$495$	0	0
$496$	1.12407e6	0.205158
$497$	−1.05741e6	−0.192023
$498$	0	0
$499$	−8.63432e6	−1.55230	−0.776152	−	0.630546i	$-0.782831\pi$
$499$	−8.63432e6	−1.55230	−0.776152	+	0.630546i	$0.782831\pi$
$500$	−2.89802e6	−0.518414
$501$	0	0
$502$	4.73529e6	0.838663
$503$	−7.38812e6	−1.30201	−0.651005	−	0.759074i	$-0.725652\pi$
$503$	−7.38812e6	−1.30201	−0.651005	+	0.759074i	$0.725652\pi$
$504$	0	0
$505$	−1.59345e7	−2.78042
$506$	−1.35801e6	−0.235790
$507$	0	0
$508$	3.68162e6	0.632965
$509$	−540033.	−0.0923902	−0.0461951	−	0.998932i	$-0.514710\pi$
$509$	−540033.	−0.0923902	−0.0461951	+	0.998932i	$0.514710\pi$
$510$	0	0
$511$	341246.	0.0578116
$512$	262144.	0.0441942
$513$	0	0
$514$	−5.14715e6	−0.859328
$515$	−8.59382e6	−1.42780
$516$	0	0
$517$	302712.	0.0498084
$518$	61131.8	0.0100102
$519$	0	0
$520$	−5.68296e6	−0.921651
$521$	1.12129e6	0.180978	0.0904889	−	0.995897i	$-0.471157\pi$
$521$	1.12129e6	0.180978	0.0904889	+	0.995897i	$0.471157\pi$
$522$	0	0
$523$	−6.91762e6	−1.10587	−0.552933	−	0.833226i	$-0.686491\pi$
$523$	−6.91762e6	−1.10587	−0.552933	+	0.833226i	$0.686491\pi$
$524$	4.39421e6	0.699122
$525$	0	0
$526$	3.83961e6	0.605094
$527$	2.79101e6	0.437758
$528$	0	0
$529$	1.84422e7	2.86533
$530$	−9.91626e6	−1.53341
$531$	0	0
$532$	−699515.	−0.107156
$533$	83285.3	0.0126984
$534$	0	0
$535$	4.11618e6	0.621741
$536$	119287.	0.0179342
$537$	0	0
$538$	−3.69526e6	−0.550413
$539$	−1.11865e6	−0.165853
$540$	0	0
$541$	−8.25073e6	−1.21199	−0.605995	−	0.795468i	$-0.707225\pi$
$541$	−8.25073e6	−1.21199	−0.605995	+	0.795468i	$0.707225\pi$
$542$	−3.16141e6	−0.462256
$543$	0	0
$544$	650889.	0.0942996
$545$	5.06786e6	0.730859
$546$	0	0
$547$	−1.85242e6	−0.264710	−0.132355	−	0.991202i	$-0.542254\pi$
$547$	−1.85242e6	−0.264710	−0.132355	+	0.991202i	$0.542254\pi$
$548$	−4.17184e6	−0.593439
$549$	0	0
$550$	1.39392e6	0.196486
$551$	6.66214e6	0.934834
$552$	0	0
$553$	1.34474e6	0.186993
$554$	−425951.	−0.0589638
$555$	0	0
$556$	6.81968e6	0.935572
$557$	−6.80130e6	−0.928868	−0.464434	−	0.885608i	$-0.653742\pi$
$557$	−6.80130e6	−0.928868	−0.464434	+	0.885608i	$0.653742\pi$
$558$	0	0
$559$	5.52040e6	0.747208
$560$	−448420.	−0.0604247
$561$	0	0
$562$	7.12902e6	0.952115
$563$	1.09352e7	1.45397	0.726983	−	0.686655i	$-0.240922\pi$
$563$	1.09352e7	1.45397	0.726983	+	0.686655i	$0.240922\pi$
$564$	0	0
$565$	1.19339e7	1.57276
$566$	7.42070e6	0.973652
$567$	0	0
$568$	−3.50808e6	−0.456246
$569$	8.01933e6	1.03838	0.519191	−	0.854658i	$-0.326233\pi$
$569$	8.01933e6	1.03838	0.519191	+	0.854658i	$0.326233\pi$
$570$	0	0
$571$	−3.02983e6	−0.388891	−0.194445	−	0.980913i	$-0.562291\pi$
$571$	−3.02983e6	−0.388891	−0.194445	+	0.980913i	$0.562291\pi$
$572$	1.06501e6	0.136102
$573$	0	0
$574$	6571.71	0.000832527 0
$575$	−2.55366e7	−3.22102
$576$	0	0
$577$	−1.40643e7	−1.75864	−0.879321	−	0.476230i	$-0.842003\pi$
$577$	−1.40643e7	−1.75864	−0.879321	+	0.476230i	$0.842003\pi$
$578$	−4.06330e6	−0.505894
$579$	0	0
$580$	4.27072e6	0.527146
$581$	−438346.	−0.0538737
$582$	0	0
$583$	1.85835e6	0.226441
$584$	1.13212e6	0.137360
$585$	0	0
$586$	−4.99673e6	−0.601093
$587$	−3.16272e6	−0.378848	−0.189424	−	0.981895i	$-0.560662\pi$
$587$	−3.16272e6	−0.378848	−0.189424	+	0.981895i	$0.560662\pi$
$588$	0	0
$589$	−9.95120e6	−1.18192
$590$	−1.26431e6	−0.149528
$591$	0	0
$592$	202811.	0.0237841
$593$	5.09693e6	0.595213	0.297606	−	0.954689i	$-0.403812\pi$
$593$	5.09693e6	0.595213	0.297606	+	0.954689i	$0.403812\pi$
$594$	0	0
$595$	−1.11340e6	−0.128932
$596$	4.17328e6	0.481240
$597$	0	0
$598$	−1.95109e7	−2.23113
$599$	1.57513e7	1.79370	0.896851	−	0.442334i	$-0.145849\pi$
$599$	1.57513e7	1.79370	0.896851	+	0.442334i	$0.145849\pi$
$600$	0	0
$601$	−6.38437e6	−0.720994	−0.360497	−	0.932760i	$-0.617393\pi$
$601$	−6.38437e6	−0.720994	−0.360497	+	0.932760i	$0.617393\pi$
$602$	435593.	0.0489880
$603$	0	0
$604$	4.90049e6	0.546572
$605$	1.42029e7	1.57757
$606$	0	0
$607$	−1.45247e6	−0.160006	−0.0800028	−	0.996795i	$-0.525493\pi$
$607$	−1.45247e6	−0.160006	−0.0800028	+	0.996795i	$0.525493\pi$
$608$	−2.32071e6	−0.254603
$609$	0	0
$610$	1.96425e7	2.13734
$611$	4.34917e6	0.471306
$612$	0	0
$613$	−1.11755e7	−1.20121	−0.600603	−	0.799547i	$-0.705073\pi$
$613$	−1.11755e7	−1.20121	−0.600603	+	0.799547i	$0.705073\pi$
$614$	−8.32413e6	−0.891083
$615$	0	0
$616$	84035.8	0.00892303
$617$	−3.29875e6	−0.348848	−0.174424	−	0.984671i	$-0.555806\pi$
$617$	−3.29875e6	−0.348848	−0.174424	+	0.984671i	$0.555806\pi$
$618$	0	0
$619$	1.13187e7	1.18732	0.593661	−	0.804715i	$-0.297682\pi$
$619$	1.13187e7	1.18732	0.593661	+	0.804715i	$0.297682\pi$
$620$	−6.37915e6	−0.666475
$621$	0	0
$622$	−9.78271e6	−1.01387
$623$	987164.	0.101899
$624$	0	0
$625$	447120.	0.0457851
$626$	1.25782e7	1.28287
$627$	0	0
$628$	−4.02095e6	−0.406846
$629$	503569.	0.0507496
$630$	0	0
$631$	1.14284e6	0.114264	0.0571322	−	0.998367i	$-0.481804\pi$
$631$	1.14284e6	0.114264	0.0571322	+	0.998367i	$0.481804\pi$
$632$	4.46132e6	0.444294
$633$	0	0
$634$	5.89283e6	0.582238
$635$	−2.08934e7	−2.05624
$636$	0	0
$637$	−1.60721e7	−1.56936
$638$	−800351.	−0.0778447
$639$	0	0
$640$	−1.48768e6	−0.143569
$641$	2.76850e6	0.266133	0.133067	−	0.991107i	$-0.457518\pi$
$641$	2.76850e6	0.266133	0.133067	+	0.991107i	$0.457518\pi$
$642$	0	0
$643$	2.93152e6	0.279618	0.139809	−	0.990178i	$-0.455351\pi$
$643$	2.93152e6	0.279618	0.139809	+	0.990178i	$0.455351\pi$
$644$	−1.53953e6	−0.146276
$645$	0	0
$646$	−5.76221e6	−0.543260
$647$	−1.24967e7	−1.17364	−0.586822	−	0.809716i	$-0.699621\pi$
$647$	−1.24967e7	−1.17364	−0.586822	+	0.809716i	$0.699621\pi$
$648$	0	0
$649$	236937.	0.0220811
$650$	2.00270e7	1.85923
$651$	0	0
$652$	2.26172e6	0.208363
$653$	1.51758e7	1.39273	0.696367	−	0.717686i	$-0.254798\pi$
$653$	1.51758e7	1.39273	0.696367	+	0.717686i	$0.254798\pi$
$654$	0	0
$655$	−2.49374e7	−2.27116
$656$	21802.3	0.00197808
$657$	0	0
$658$	343175.	0.0308995
$659$	5.13325e6	0.460446	0.230223	−	0.973138i	$-0.426054\pi$
$659$	5.13325e6	0.460446	0.230223	+	0.973138i	$0.426054\pi$
$660$	0	0
$661$	3.04537e6	0.271104	0.135552	−	0.990770i	$-0.456719\pi$
$661$	3.04537e6	0.271104	0.135552	+	0.990770i	$0.456719\pi$
$662$	−1.04622e7	−0.927850
$663$	0	0
$664$	−1.45426e6	−0.128003
$665$	3.96978e6	0.348107
$666$	0	0
$667$	1.46624e7	1.27612
$668$	−649649.	−0.0563297
$669$	0	0
$670$	−676959.	−0.0582606
$671$	−3.68110e6	−0.315625
$672$	0	0
$673$	−7.81158e6	−0.664815	−0.332408	−	0.943136i	$-0.607861\pi$
$673$	−7.81158e6	−0.664815	−0.332408	+	0.943136i	$0.607861\pi$
$674$	−5.25527e6	−0.445600
$675$	0	0
$676$	9.36071e6	0.787847
$677$	6.52434e6	0.547097	0.273549	−	0.961858i	$-0.411803\pi$
$677$	6.52434e6	0.547097	0.273549	+	0.961858i	$0.411803\pi$
$678$	0	0
$679$	3.21878e6	0.267927
$680$	−3.69383e6	−0.306340
$681$	0	0
$682$	1.19548e6	0.0984196
$683$	−1.98593e7	−1.62896	−0.814482	−	0.580189i	$-0.802979\pi$
$683$	−1.98593e7	−1.62896	−0.814482	+	0.580189i	$0.802979\pi$
$684$	0	0
$685$	2.36754e7	1.92784
$686$	−2.56508e6	−0.208109
$687$	0	0
$688$	1.44512e6	0.116395
$689$	2.66996e7	2.14267
$690$	0	0
$691$	1.73704e7	1.38393	0.691965	−	0.721931i	$-0.256745\pi$
$691$	1.73704e7	1.38393	0.691965	+	0.721931i	$0.256745\pi$
$692$	1.75685e6	0.139466
$693$	0	0
$694$	8.61828e6	0.679238
$695$	−3.87020e7	−3.03928
$696$	0	0
$697$	54134.0	0.00422073
$698$	−489259.	−0.0380102
$699$	0	0
$700$	1.58025e6	0.121894
$701$	2.35091e7	1.80693	0.903466	−	0.428660i	$-0.141014\pi$
$701$	2.35091e7	1.80693	0.903466	+	0.428660i	$0.141014\pi$
$702$	0	0
$703$	−1.79545e6	−0.137020
$704$	278797.	0.0212010
$705$	0	0
$706$	6.11285e6	0.461564
$707$	3.38536e6	0.254716
$708$	0	0
$709$	−1.57935e7	−1.17995	−0.589975	−	0.807421i	$-0.700863\pi$
$709$	−1.57935e7	−1.17995	−0.589975	+	0.807421i	$0.700863\pi$
$710$	1.99085e7	1.48215
$711$	0	0
$712$	3.27502e6	0.242111
$713$	−2.19011e7	−1.61340
$714$	0	0
$715$	−6.04399e6	−0.442139
$716$	4.77150e6	0.347834
$717$	0	0
$718$	−2.28531e6	−0.165437
$719$	−9.21582e6	−0.664832	−0.332416	−	0.943133i	$-0.607864\pi$
$719$	−9.21582e6	−0.664832	−0.332416	+	0.943133i	$0.607864\pi$
$720$	0	0
$721$	1.82580e6	0.130802
$722$	1.06405e7	0.759658
$723$	0	0
$724$	4.67350e6	0.331356
$725$	−1.50502e7	−1.06340
$726$	0	0
$727$	−915909.	−0.0642712	−0.0321356	−	0.999484i	$-0.510231\pi$
$727$	−915909.	−0.0642712	−0.0321356	+	0.999484i	$0.510231\pi$
$728$	1.20737e6	0.0844331
$729$	0	0
$730$	−6.42482e6	−0.446225
$731$	3.58816e6	0.248358
$732$	0	0
$733$	−7.75153e6	−0.532878	−0.266439	−	0.963852i	$-0.585847\pi$
$733$	−7.75153e6	−0.532878	−0.266439	+	0.963852i	$0.585847\pi$
$734$	−1.89767e7	−1.30011
$735$	0	0
$736$	−5.10755e6	−0.347551
$737$	126865.	0.00860346
$738$	0	0
$739$	1.76153e7	1.18653	0.593267	−	0.805006i	$-0.297838\pi$
$739$	1.76153e7	1.18653	0.593267	+	0.805006i	$0.297838\pi$
$740$	−1.15096e6	−0.0772648
$741$	0	0
$742$	2.10675e6	0.140477
$743$	4.18084e6	0.277838	0.138919	−	0.990304i	$-0.455637\pi$
$743$	4.18084e6	0.277838	0.138919	+	0.990304i	$0.455637\pi$
$744$	0	0
$745$	−2.36835e7	−1.56335
$746$	1.04840e7	0.689729
$747$	0	0
$748$	692239.	0.0452379
$749$	−874501.	−0.0569581
$750$	0	0
$751$	−201977.	−0.0130678	−0.00653389	−	0.999979i	$-0.502080\pi$
$751$	−201977.	−0.0130678	−0.00653389	+	0.999979i	$0.502080\pi$
$752$	1.13852e6	0.0734169
$753$	0	0
$754$	−1.14989e7	−0.736596
$755$	−2.78105e7	−1.77558
$756$	0	0
$757$	8.26507e6	0.524212	0.262106	−	0.965039i	$-0.415583\pi$
$757$	8.26507e6	0.524212	0.262106	+	0.965039i	$0.415583\pi$
$758$	161985.	0.0102401
$759$	0	0
$760$	1.31702e7	0.827098
$761$	−2.87626e7	−1.80039	−0.900195	−	0.435487i	$-0.856576\pi$
$761$	−2.87626e7	−1.80039	−0.900195	+	0.435487i	$0.856576\pi$
$762$	0	0
$763$	−1.07669e6	−0.0669545
$764$	6.93231e6	0.429679
$765$	0	0
$766$	7.45634e6	0.459149
$767$	3.40416e6	0.208940
$768$	0	0
$769$	4.49101e6	0.273860	0.136930	−	0.990581i	$-0.456276\pi$
$769$	4.49101e6	0.273860	0.136930	+	0.990581i	$0.456276\pi$
$770$	−476907.	−0.0289872
$771$	0	0
$772$	1.54390e6	0.0932344
$773$	2.82516e7	1.70057	0.850286	−	0.526322i	$-0.176429\pi$
$773$	2.82516e7	1.70057	0.850286	+	0.526322i	$0.176429\pi$
$774$	0	0
$775$	2.24804e7	1.34447
$776$	1.06786e7	0.636593
$777$	0	0
$778$	−1.50379e7	−0.890715
$779$	−193012.	−0.0113957
$780$	0	0
$781$	−3.73094e6	−0.218872
$782$	−1.26818e7	−0.741589
$783$	0	0
$784$	−4.20732e6	−0.244464
$785$	2.28191e7	1.32167
$786$	0	0
$787$	−1.40642e6	−0.0809428	−0.0404714	−	0.999181i	$-0.512886\pi$
$787$	−1.40642e6	−0.0809428	−0.0404714	+	0.999181i	$0.512886\pi$
$788$	1.34807e7	0.773385
$789$	0	0
$790$	−2.53182e7	−1.44333
$791$	−2.53541e6	−0.144081
$792$	0	0
$793$	−5.28876e7	−2.98656
$794$	−1.86967e7	−1.05248
$795$	0	0
$796$	1.37722e7	0.770410
$797$	2.68656e7	1.49814	0.749068	−	0.662494i	$-0.230502\pi$
$797$	2.68656e7	1.49814	0.749068	+	0.662494i	$0.230502\pi$
$798$	0	0
$799$	2.82688e6	0.156654
$800$	5.24264e6	0.289618
$801$	0	0
$802$	1.12668e7	0.618538
$803$	1.20404e6	0.0658949
$804$	0	0
$805$	8.73690e6	0.475190
$806$	1.71759e7	0.931283
$807$	0	0
$808$	1.12313e7	0.605203
$809$	−5.91282e6	−0.317632	−0.158816	−	0.987308i	$-0.550768\pi$
$809$	−5.91282e6	−0.317632	−0.158816	+	0.987308i	$0.550768\pi$
$810$	0	0
$811$	−3.14466e7	−1.67889	−0.839443	−	0.543448i	$-0.817119\pi$
$811$	−3.14466e7	−1.67889	−0.839443	+	0.543448i	$0.817119\pi$
$812$	−907334.	−0.0482922
$813$	0	0
$814$	215695.	0.0114098
$815$	−1.28354e7	−0.676885
$816$	0	0
$817$	−1.27934e7	−0.670551
$818$	1.44530e7	0.755224
$819$	0	0
$820$	−123729.	−0.00642595
$821$	1.26936e6	0.0657246	0.0328623	−	0.999460i	$-0.489538\pi$
$821$	1.26936e6	0.0657246	0.0328623	+	0.999460i	$0.489538\pi$
$822$	0	0
$823$	−1.45973e7	−0.751228	−0.375614	−	0.926776i	$-0.622568\pi$
$823$	−1.45973e7	−0.751228	−0.375614	+	0.926776i	$0.622568\pi$
$824$	6.05727e6	0.310784
$825$	0	0
$826$	268608.	0.0136984
$827$	857506.	0.0435987	0.0217993	−	0.999762i	$-0.493061\pi$
$827$	857506.	0.0435987	0.0217993	+	0.999762i	$0.493061\pi$
$828$	0	0
$829$	−3.05043e6	−0.154161	−0.0770804	−	0.997025i	$-0.524560\pi$
$829$	−3.05043e6	−0.154161	−0.0770804	+	0.997025i	$0.524560\pi$
$830$	8.25298e6	0.415830
$831$	0	0
$832$	4.00558e6	0.200612
$833$	−1.04466e7	−0.521628
$834$	0	0
$835$	3.68679e6	0.182992
$836$	−2.46814e6	−0.122139
$837$	0	0
$838$	8.93538e6	0.439545
$839$	107056.	0.00525059	0.00262529	−	0.999997i	$-0.499164\pi$
$839$	107056.	0.00525059	0.00262529	+	0.999997i	$0.499164\pi$
$840$	0	0
$841$	−1.18698e7	−0.578698
$842$	2.43526e7	1.18376
$843$	0	0
$844$	−3.74389e6	−0.180912
$845$	−5.31225e7	−2.55939
$846$	0	0
$847$	−3.01747e6	−0.144522
$848$	6.98938e6	0.333771
$849$	0	0
$850$	1.30172e7	0.617974
$851$	−3.95152e6	−0.187043
$852$	0	0
$853$	−2.31261e7	−1.08825	−0.544126	−	0.839003i	$-0.683139\pi$
$853$	−2.31261e7	−1.08825	−0.544126	+	0.839003i	$0.683139\pi$
$854$	−4.17315e6	−0.195803
$855$	0	0
$856$	−2.90125e6	−0.135332
$857$	1.12261e7	0.522129	0.261064	−	0.965321i	$-0.415927\pi$
$857$	1.12261e7	0.522129	0.261064	+	0.965321i	$0.415927\pi$
$858$	0	0
$859$	−8.10257e6	−0.374662	−0.187331	−	0.982297i	$-0.559984\pi$
$859$	−8.10257e6	−0.374662	−0.187331	+	0.982297i	$0.559984\pi$
$860$	−8.20114e6	−0.378119
$861$	0	0
$862$	2.50742e7	1.14937
$863$	1.98271e6	0.0906217	0.0453109	−	0.998973i	$-0.485572\pi$
$863$	1.98271e6	0.0906217	0.0453109	+	0.998973i	$0.485572\pi$
$864$	0	0
$865$	−9.97020e6	−0.453068
$866$	−5.89183e6	−0.266966
$867$	0	0
$868$	1.35528e6	0.0610562
$869$	4.74474e6	0.213139
$870$	0	0
$871$	1.82271e6	0.0814092
$872$	−3.57204e6	−0.159083
$873$	0	0
$874$	4.52162e7	2.00224
$875$	−3.49411e6	−0.154283
$876$	0	0
$877$	7.19813e6	0.316025	0.158012	−	0.987437i	$-0.449491\pi$
$877$	7.19813e6	0.316025	0.158012	+	0.987437i	$0.449491\pi$
$878$	−1.15395e7	−0.505184
$879$	0	0
$880$	−1.58219e6	−0.0688734
$881$	−3.75441e6	−0.162968	−0.0814840	−	0.996675i	$-0.525966\pi$
$881$	−3.75441e6	−0.162968	−0.0814840	+	0.996675i	$0.525966\pi$
$882$	0	0
$883$	4.48940e7	1.93770	0.968850	−	0.247649i	$-0.0796581\pi$
$883$	4.48940e7	1.93770	0.968850	+	0.247649i	$0.0796581\pi$
$884$	9.94564e6	0.428058
$885$	0	0
$886$	−2.51526e6	−0.107646
$887$	4.95164e6	0.211320	0.105660	−	0.994402i	$-0.466305\pi$
$887$	4.95164e6	0.211320	0.105660	+	0.994402i	$0.466305\pi$
$888$	0	0
$889$	4.43889e6	0.188374
$890$	−1.85859e7	−0.786517
$891$	0	0
$892$	543646.	0.0228773
$893$	−1.00791e7	−0.422955
$894$	0	0
$895$	−2.70785e7	−1.12997
$896$	316064.	0.0131524
$897$	0	0
$898$	2.45778e7	1.01707
$899$	−1.29076e7	−0.532655
$900$	0	0
$901$	1.73542e7	0.712187
$902$	23187.4	0.000948932 0
$903$	0	0
$904$	−8.41149e6	−0.342336
$905$	−2.65223e7	−1.07644
$906$	0	0
$907$	3.76360e7	1.51910	0.759549	−	0.650451i	$-0.225420\pi$
$907$	3.76360e7	1.51910	0.759549	+	0.650451i	$0.225420\pi$
$908$	−448058.	−0.0180351
$909$	0	0
$910$	−6.85189e6	−0.274288
$911$	2.13988e7	0.854267	0.427134	−	0.904189i	$-0.359523\pi$
$911$	2.13988e7	0.854267	0.427134	+	0.904189i	$0.359523\pi$
$912$	0	0
$913$	−1.54664e6	−0.0614064
$914$	82877.4	0.00328149
$915$	0	0
$916$	−2.21954e7	−0.874026
$917$	5.29806e6	0.208062
$918$	0	0
$919$	1.55931e7	0.609038	0.304519	−	0.952506i	$-0.401504\pi$
$919$	1.55931e7	0.609038	0.304519	+	0.952506i	$0.401504\pi$
$920$	2.89856e7	1.12905
$921$	0	0
$922$	−1.68742e7	−0.653727
$923$	−5.36038e7	−2.07105
$924$	0	0
$925$	4.05604e6	0.155865
$926$	5.52364e6	0.211689
$927$	0	0
$928$	−3.01018e6	−0.114742
$929$	−2.51446e7	−0.955884	−0.477942	−	0.878392i	$-0.658617\pi$
$929$	−2.51446e7	−0.955884	−0.477942	+	0.878392i	$0.658617\pi$
$930$	0	0
$931$	3.72467e7	1.40836
$932$	796934.	0.0300526
$933$	0	0
$934$	1.49147e7	0.559433
$935$	−3.92849e6	−0.146959
$936$	0	0
$937$	8.15147e6	0.303310	0.151655	−	0.988433i	$-0.451540\pi$
$937$	8.15147e6	0.303310	0.151655	+	0.988433i	$0.451540\pi$
$938$	143823.	0.00533730
$939$	0	0
$940$	−6.46115e6	−0.238501
$941$	1.00929e7	0.371570	0.185785	−	0.982590i	$-0.440517\pi$
$941$	1.00929e7	0.371570	0.185785	+	0.982590i	$0.440517\pi$
$942$	0	0
$943$	−424791.	−0.0155559
$944$	891136.	0.0325472
$945$	0	0
$946$	1.53693e6	0.0558375
$947$	−2.02114e7	−0.732356	−0.366178	−	0.930545i	$-0.619334\pi$
$947$	−2.02114e7	−0.732356	−0.366178	+	0.930545i	$0.619334\pi$
$948$	0	0
$949$	1.72989e7	0.623522
$950$	−4.64122e7	−1.66849
$951$	0	0
$952$	784770.	0.0280640
$953$	−2.40883e7	−0.859159	−0.429580	−	0.903029i	$-0.641338\pi$
$953$	−2.40883e7	−0.859159	−0.429580	+	0.903029i	$0.641338\pi$
$954$	0	0
$955$	−3.93412e7	−1.39585
$956$	4.26543e6	0.150945
$957$	0	0
$958$	3.57733e6	0.125935
$959$	−5.02995e6	−0.176611
$960$	0	0
$961$	−9.34914e6	−0.326560
$962$	3.09897e6	0.107964
$963$	0	0
$964$	1.68064e7	0.582482
$965$	−8.76170e6	−0.302880
$966$	0	0
$967$	3.45319e6	0.118756	0.0593778	−	0.998236i	$-0.481088\pi$
$967$	3.45319e6	0.118756	0.0593778	+	0.998236i	$0.481088\pi$
$968$	−1.00108e7	−0.343383
$969$	0	0
$970$	−6.06018e7	−2.06803
$971$	−2.59394e7	−0.882901	−0.441451	−	0.897286i	$-0.645536\pi$
$971$	−2.59394e7	−0.882901	−0.441451	+	0.897286i	$0.645536\pi$
$972$	0	0
$973$	8.22242e6	0.278431
$974$	2.81987e7	0.952428
$975$	0	0
$976$	−1.38449e7	−0.465226
$977$	1.13153e7	0.379253	0.189626	−	0.981856i	$-0.439272\pi$
$977$	1.13153e7	0.379253	0.189626	+	0.981856i	$0.439272\pi$
$978$	0	0
$979$	3.48307e6	0.116146
$980$	2.38767e7	0.794163
$981$	0	0
$982$	7.80824e6	0.258389
$983$	5.59419e7	1.84652	0.923259	−	0.384179i	$-0.125516\pi$
$983$	5.59419e7	1.84652	0.923259	+	0.384179i	$0.125516\pi$
$984$	0	0
$985$	−7.65033e7	−2.51241
$986$	−7.47410e6	−0.244831
$987$	0	0
$988$	−3.54607e7	−1.15573
$989$	−2.81564e7	−0.915350
$990$	0	0
$991$	−6.00724e7	−1.94308	−0.971540	−	0.236875i	$-0.923877\pi$
$991$	−6.00724e7	−1.94308	−0.971540	+	0.236875i	$0.923877\pi$
$992$	4.49628e6	0.145069
$993$	0	0
$994$	−4.22966e6	−0.135781
$995$	−7.81581e7	−2.50274
$996$	0	0
$997$	−1.94667e7	−0.620232	−0.310116	−	0.950699i	$-0.600368\pi$
$997$	−1.94667e7	−0.620232	−0.310116	+	0.950699i	$0.600368\pi$
$998$	−3.45373e7	−1.09765
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1062.6.a.m.1.1		8
3.2	odd	2		354.6.a.h.1.8	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
354.6.a.h.1.8	✓	8	3.2	odd	2
1062.6.a.m.1.1		8	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 1062 = 2 \cdot 3^{2} \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	1062.a (trivial)

\(f(q)\)	\(=\)	\(q+4.00000 q^{2} +16.0000 q^{4} -90.8007 q^{5} +19.2910 q^{7} +64.0000 q^{8} +O(q^{10})\)	\(q+4.00000 q^{2} +16.0000 q^{4} -90.8007 q^{5} +19.2910 q^{7} +64.0000 q^{8} -363.203 q^{10} +68.0658 q^{11} +977.925 q^{13} +77.1641 q^{14} +256.000 q^{16} +635.634 q^{17} -2266.32 q^{19} -1452.81 q^{20} +272.263 q^{22} -4987.84 q^{23} +5119.77 q^{25} +3911.70 q^{26} +308.657 q^{28} -2939.63 q^{29} +4390.90 q^{31} +1024.00 q^{32} +2542.54 q^{34} -1751.64 q^{35} +792.231 q^{37} -9065.29 q^{38} -5811.24 q^{40} +85.1653 q^{41} +5645.02 q^{43} +1089.05 q^{44} -19951.4 q^{46} +4447.34 q^{47} -16434.9 q^{49} +20479.1 q^{50} +15646.8 q^{52} +27302.3 q^{53} -6180.42 q^{55} +1234.63 q^{56} -11758.5 q^{58} +3481.00 q^{59} -54081.5 q^{61} +17563.6 q^{62} +4096.00 q^{64} -88796.3 q^{65} +1863.86 q^{67} +10170.1 q^{68} -7006.56 q^{70} -54813.8 q^{71} +17689.4 q^{73} +3168.92 q^{74} -36261.2 q^{76} +1313.06 q^{77} +69708.2 q^{79} -23245.0 q^{80} +340.661 q^{82} -22722.8 q^{83} -57716.0 q^{85} +22580.1 q^{86} +4356.21 q^{88} +51172.2 q^{89} +18865.2 q^{91} -79805.5 q^{92} +17789.4 q^{94} +205784. q^{95} +166854. q^{97} -65739.4 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 32 q^{2} + 128 q^{4} - 40 q^{5} + 181 q^{7} + 512 q^{8}+O(q^{10}) \) 8 * q + 32 * q^2 + 128 * q^4 - 40 * q^5 + 181 * q^7 + 512 * q^8	\( 8 q + 32 q^{2} + 128 q^{4} - 40 q^{5} + 181 q^{7} + 512 q^{8} - 160 q^{10} + 349 q^{11} + 121 q^{13} + 724 q^{14} + 2048 q^{16} - 437 q^{17} + 1314 q^{19} - 640 q^{20} + 1396 q^{22} - 1224 q^{23} + 9592 q^{25} + 484 q^{26} + 2896 q^{28} - 5276 q^{29} + 18332 q^{31} + 8192 q^{32} - 1748 q^{34} - 19518 q^{35} + 30331 q^{37} + 5256 q^{38} - 2560 q^{40} - 8323 q^{41} + 30851 q^{43} + 5584 q^{44} - 4896 q^{46} + 5730 q^{47} + 32295 q^{49} + 38368 q^{50} + 1936 q^{52} + 33524 q^{53} + 23660 q^{55} + 11584 q^{56} - 21104 q^{58} + 27848 q^{59} + 2692 q^{61} + 73328 q^{62} + 32768 q^{64} + 59892 q^{65} + 56244 q^{67} - 6992 q^{68} - 78072 q^{70} + 48473 q^{71} - 30796 q^{73} + 121324 q^{74} + 21024 q^{76} - 59683 q^{77} + 135513 q^{79} - 10240 q^{80} - 33292 q^{82} + 88111 q^{83} + 114418 q^{85} + 123404 q^{86} + 22336 q^{88} + 112196 q^{89} + 377433 q^{91} - 19584 q^{92} + 22920 q^{94} - 328146 q^{95} + 551378 q^{97} + 129180 q^{98}+O(q^{100}) \) 8 * q + 32 * q^2 + 128 * q^4 - 40 * q^5 + 181 * q^7 + 512 * q^8 - 160 * q^10 + 349 * q^11 + 121 * q^13 + 724 * q^14 + 2048 * q^16 - 437 * q^17 + 1314 * q^19 - 640 * q^20 + 1396 * q^22 - 1224 * q^23 + 9592 * q^25 + 484 * q^26 + 2896 * q^28 - 5276 * q^29 + 18332 * q^31 + 8192 * q^32 - 1748 * q^34 - 19518 * q^35 + 30331 * q^37 + 5256 * q^38 - 2560 * q^40 - 8323 * q^41 + 30851 * q^43 + 5584 * q^44 - 4896 * q^46 + 5730 * q^47 + 32295 * q^49 + 38368 * q^50 + 1936 * q^52 + 33524 * q^53 + 23660 * q^55 + 11584 * q^56 - 21104 * q^58 + 27848 * q^59 + 2692 * q^61 + 73328 * q^62 + 32768 * q^64 + 59892 * q^65 + 56244 * q^67 - 6992 * q^68 - 78072 * q^70 + 48473 * q^71 - 30796 * q^73 + 121324 * q^74 + 21024 * q^76 - 59683 * q^77 + 135513 * q^79 - 10240 * q^80 - 33292 * q^82 + 88111 * q^83 + 114418 * q^85 + 123404 * q^86 + 22336 * q^88 + 112196 * q^89 + 377433 * q^91 - 19584 * q^92 + 22920 * q^94 - 328146 * q^95 + 551378 * q^97 + 129180 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more