LMFDB - Embedded newform 550.6.a.h.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [550,6,Mod(1,550)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(550, 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("550.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$N$	$=$	$550 = 2 \cdot 5^{2} \cdot 11$
Weight:	$k$	$=$	$6$
Character orbit:	$[\chi]$	$=$	550.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:	$88.2111008971$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{793})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{2} - x - 198$ x^2 - x - 198
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 22)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$14.5801$ of defining polynomial
Character	$\chi$	$=$	550.1

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-4.00000 q^{2} -28.5801 q^{3} +16.0000 q^{4} +114.321 q^{6} +91.4808 q^{7} -64.0000 q^{8} +573.824 q^{9} -121.000 q^{11} -457.282 q^{12} +463.801 q^{13} -365.923 q^{14} +256.000 q^{16} -1641.94 q^{17} -2295.29 q^{18} -1693.53 q^{19} -2614.53 q^{21} +484.000 q^{22} -75.9391 q^{23} +1829.13 q^{24} -1855.21 q^{26} -9454.98 q^{27} +1463.69 q^{28} -2942.33 q^{29} +8443.63 q^{31} -1024.00 q^{32} +3458.20 q^{33} +6567.74 q^{34} +9181.18 q^{36} +35.5610 q^{37} +6774.10 q^{38} -13255.5 q^{39} +9222.25 q^{41} +10458.1 q^{42} +11516.7 q^{43} -1936.00 q^{44} +303.756 q^{46} -6179.00 q^{47} -7316.51 q^{48} -8438.27 q^{49} +46926.7 q^{51} +7420.82 q^{52} -25255.1 q^{53} +37819.9 q^{54} -5854.77 q^{56} +48401.2 q^{57} +11769.3 q^{58} +40786.8 q^{59} -7368.85 q^{61} -33774.5 q^{62} +52493.8 q^{63} +4096.00 q^{64} -13832.8 q^{66} +11024.7 q^{67} -26271.0 q^{68} +2170.35 q^{69} -46964.9 q^{71} -36724.7 q^{72} +60727.5 q^{73} -142.244 q^{74} -27096.4 q^{76} -11069.2 q^{77} +53022.0 q^{78} +18386.1 q^{79} +130785. q^{81} -36889.0 q^{82} +59321.9 q^{83} -41832.5 q^{84} -46066.6 q^{86} +84092.1 q^{87} +7744.00 q^{88} +5070.71 q^{89} +42428.9 q^{91} -1215.03 q^{92} -241320. q^{93} +24716.0 q^{94} +29266.1 q^{96} +130795. q^{97} +33753.1 q^{98} -69432.7 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$2 q - 8 q^{2} - 29 q^{3} + 32 q^{4} + 116 q^{6} + 14 q^{7} - 128 q^{8} + 331 q^{9} - 242 q^{11} - 464 q^{12} + 646 q^{13} - 56 q^{14} + 512 q^{16} + 208 q^{17} - 1324 q^{18} - 2148 q^{19} - 2582 q^{21}+ \cdots - 40051 q^{99}+O(q^{100})$ 2 * q - 8 * q^2 - 29 * q^3 + 32 * q^4 + 116 * q^6 + 14 * q^7 - 128 * q^8 + 331 * q^9 - 242 * q^11 - 464 * q^12 + 646 * q^13 - 56 * q^14 + 512 * q^16 + 208 * q^17 - 1324 * q^18 - 2148 * q^19 - 2582 * q^21 + 968 * q^22 - 349 * q^23 + 1856 * q^24 - 2584 * q^26 - 9251 * q^27 + 224 * q^28 + 4422 * q^29 + 14381 * q^31 - 2048 * q^32 + 3509 * q^33 - 832 * q^34 + 5296 * q^36 + 4267 * q^37 + 8592 * q^38 - 13332 * q^39 - 10110 * q^41 + 10328 * q^42 + 9798 * q^43 - 3872 * q^44 + 1396 * q^46 - 21144 * q^47 - 7424 * q^48 - 19242 * q^49 + 46150 * q^51 + 10336 * q^52 - 39584 * q^53 + 37004 * q^54 - 896 * q^56 + 48592 * q^57 - 17688 * q^58 + 90951 * q^59 - 29550 * q^61 - 57524 * q^62 + 71308 * q^63 + 8192 * q^64 - 14036 * q^66 + 64149 * q^67 + 3328 * q^68 + 2285 * q^69 + 23583 * q^71 - 21184 * q^72 + 39058 * q^73 - 17068 * q^74 - 34368 * q^76 - 1694 * q^77 + 53328 * q^78 - 54974 * q^79 + 189706 * q^81 + 40440 * q^82 - 29986 * q^83 - 41312 * q^84 - 39192 * q^86 + 81000 * q^87 + 15488 * q^88 - 18047 * q^89 + 28312 * q^91 - 5584 * q^92 - 243813 * q^93 + 84576 * q^94 + 29696 * q^96 + 30309 * q^97 + 76968 * q^98 - 40051 * q^99

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
$3$	−28.5801	−1.83342	−0.916708	−	0.399558i	$-0.869164\pi$
$3$	−28.5801	−1.83342	−0.916708	+	0.399558i	$0.869164\pi$
$4$	16.0000	0.500000
$5$	0	0
$5$	0	0
$6$	114.321	1.29642
$7$	91.4808	0.705642	0.352821	−	0.935691i	$-0.385222\pi$
$7$	91.4808	0.705642	0.352821	+	0.935691i	$0.385222\pi$
$8$	−64.0000	−0.353553
$9$	573.824	2.36141
$10$	0	0
$11$	−121.000	−0.301511
$11$	−121.000	−0.301511
$12$	−457.282	−0.916708
$13$	463.801	0.761156	0.380578	−	0.924749i	$-0.375725\pi$
$13$	463.801	0.761156	0.380578	+	0.924749i	$0.375725\pi$
$14$	−365.923	−0.498965
$15$	0	0
$16$	256.000	0.250000
$17$	−1641.94	−1.37795	−0.688976	−	0.724784i	$-0.741939\pi$
$17$	−1641.94	−1.37795	−0.688976	+	0.724784i	$0.741939\pi$
$18$	−2295.29	−1.66977
$19$	−1693.53	−1.07624	−0.538118	−	0.842869i	$-0.680865\pi$
$19$	−1693.53	−1.07624	−0.538118	+	0.842869i	$0.680865\pi$
$20$	0	0
$21$	−2614.53	−1.29374
$22$	484.000	0.213201
$23$	−75.9391	−0.0299327	−0.0149663	−	0.999888i	$-0.504764\pi$
$23$	−75.9391	−0.0299327	−0.0149663	+	0.999888i	$0.504764\pi$
$24$	1829.13	0.648210
$25$	0	0
$26$	−1855.21	−0.538218
$27$	−9454.98	−2.49604
$28$	1463.69	0.352821
$29$	−2942.33	−0.649675	−0.324837	−	0.945770i	$-0.605310\pi$
$29$	−2942.33	−0.649675	−0.324837	+	0.945770i	$0.605310\pi$
$30$	0	0
$31$	8443.63	1.57807	0.789033	−	0.614351i	$-0.210582\pi$
$31$	8443.63	1.57807	0.789033	+	0.614351i	$0.210582\pi$
$32$	−1024.00	−0.176777
$33$	3458.20	0.552796
$34$	6567.74	0.974359
$35$	0	0
$36$	9181.18	1.18071
$37$	35.5610	0.00427041	0.00213520	−	0.999998i	$-0.499320\pi$
$37$	35.5610	0.00427041	0.00213520	+	0.999998i	$0.499320\pi$
$38$	6774.10	0.761014
$39$	−13255.5	−1.39552
$40$	0	0
$41$	9222.25	0.856796	0.428398	−	0.903590i	$-0.359078\pi$
$41$	9222.25	0.856796	0.428398	+	0.903590i	$0.359078\pi$
$42$	10458.1	0.914810
$43$	11516.7	0.949851	0.474925	−	0.880026i	$-0.342475\pi$
$43$	11516.7	0.949851	0.474925	+	0.880026i	$0.342475\pi$
$44$	−1936.00	−0.150756
$45$	0	0
$46$	303.756	0.0211656
$47$	−6179.00	−0.408013	−0.204006	−	0.978970i	$-0.565396\pi$
$47$	−6179.00	−0.408013	−0.204006	+	0.978970i	$0.565396\pi$
$48$	−7316.51	−0.458354
$49$	−8438.27	−0.502069
$50$	0	0
$51$	46926.7	2.52636
$52$	7420.82	0.380578
$53$	−25255.1	−1.23498	−0.617489	−	0.786579i	$-0.711850\pi$
$53$	−25255.1	−1.23498	−0.617489	+	0.786579i	$0.711850\pi$
$54$	37819.9	1.76497
$55$	0	0
$56$	−5854.77	−0.249482
$57$	48401.2	1.97319
$58$	11769.3	0.459389
$59$	40786.8	1.52542	0.762711	−	0.646740i	$-0.223868\pi$
$59$	40786.8	1.52542	0.762711	+	0.646740i	$0.223868\pi$
$60$	0	0
$61$	−7368.85	−0.253557	−0.126778	−	0.991931i	$-0.540464\pi$
$61$	−7368.85	−0.253557	−0.126778	+	0.991931i	$0.540464\pi$
$62$	−33774.5	−1.11586
$63$	52493.8	1.66631
$64$	4096.00	0.125000
$65$	0	0
$66$	−13832.8	−0.390886
$67$	11024.7	0.300041	0.150020	−	0.988683i	$-0.452066\pi$
$67$	11024.7	0.300041	0.150020	+	0.988683i	$0.452066\pi$
$68$	−26271.0	−0.688976
$69$	2170.35	0.0548791
$70$	0	0
$71$	−46964.9	−1.10567	−0.552837	−	0.833289i	$-0.686455\pi$
$71$	−46964.9	−1.10567	−0.552837	+	0.833289i	$0.686455\pi$
$72$	−36724.7	−0.834886
$73$	60727.5	1.33376	0.666880	−	0.745165i	$-0.267629\pi$
$73$	60727.5	1.33376	0.666880	+	0.745165i	$0.267629\pi$
$74$	−142.244	−0.00301963
$75$	0	0
$76$	−27096.4	−0.538118
$77$	−11069.2	−0.212759
$78$	53022.0	0.986778
$79$	18386.1	0.331452	0.165726	−	0.986172i	$-0.447003\pi$
$79$	18386.1	0.331452	0.165726	+	0.986172i	$0.447003\pi$
$80$	0	0
$81$	130785.	2.21486
$82$	−36889.0	−0.605846
$83$	59321.9	0.945192	0.472596	−	0.881279i	$-0.343317\pi$
$83$	59321.9	0.945192	0.472596	+	0.881279i	$0.343317\pi$
$84$	−41832.5	−0.646868
$85$	0	0
$86$	−46066.6	−0.671646
$87$	84092.1	1.19112
$88$	7744.00	0.106600
$89$	5070.71	0.0678568	0.0339284	−	0.999424i	$-0.489198\pi$
$89$	5070.71	0.0678568	0.0339284	+	0.999424i	$0.489198\pi$
$90$	0	0
$91$	42428.9	0.537104
$92$	−1215.03	−0.0149663
$93$	−241320.	−2.89325
$94$	24716.0	0.288508
$95$	0	0
$96$	29266.1	0.324105
$97$	130795.	1.41143	0.705717	−	0.708494i	$-0.250625\pi$
$97$	130795.	1.41143	0.705717	+	0.708494i	$0.250625\pi$
$98$	33753.1	0.355016
$99$	−69432.7	−0.711993
$100$	0	0
$101$	−27114.8	−0.264486	−0.132243	−	0.991217i	$-0.542218\pi$
$101$	−27114.8	−0.264486	−0.132243	+	0.991217i	$0.542218\pi$
$102$	−187707.	−1.78640
$103$	12802.9	0.118909	0.0594546	−	0.998231i	$-0.481064\pi$
$103$	12802.9	0.118909	0.0594546	+	0.998231i	$0.481064\pi$
$104$	−29683.3	−0.269109
$105$	0	0
$106$	101020.	0.873261
$107$	64131.5	0.541516	0.270758	−	0.962647i	$-0.412726\pi$
$107$	64131.5	0.541516	0.270758	+	0.962647i	$0.412726\pi$
$108$	−151280.	−1.24802
$109$	−126630.	−1.02087	−0.510434	−	0.859917i	$-0.670515\pi$
$109$	−126630.	−1.02087	−0.510434	+	0.859917i	$0.670515\pi$
$110$	0	0
$111$	−1016.34	−0.00782943
$112$	23419.1	0.176411
$113$	−131953.	−0.972130	−0.486065	−	0.873923i	$-0.661568\pi$
$113$	−131953.	−0.972130	−0.486065	+	0.873923i	$0.661568\pi$
$114$	−193605.	−1.39526
$115$	0	0
$116$	−47077.2	−0.324837
$117$	266140.	1.79740
$118$	−163147.	−1.07864
$119$	−150206.	−0.972341
$120$	0	0
$121$	14641.0	0.0909091
$122$	29475.4	0.179292
$123$	−263573.	−1.57086
$124$	135098.	0.789033
$125$	0	0
$126$	−209975.	−1.17826
$127$	−118876.	−0.654013	−0.327006	−	0.945022i	$-0.606040\pi$
$127$	−118876.	−0.654013	−0.327006	+	0.945022i	$0.606040\pi$
$128$	−16384.0	−0.0883883
$129$	−329148.	−1.74147
$130$	0	0
$131$	205935.	1.04846	0.524229	−	0.851577i	$-0.324353\pi$
$131$	205935.	1.04846	0.524229	+	0.851577i	$0.324353\pi$
$132$	55331.1	0.276398
$133$	−154925.	−0.759438
$134$	−44098.8	−0.212161
$135$	0	0
$136$	105084.	0.487179
$137$	196568.	0.894769	0.447385	−	0.894342i	$-0.352355\pi$
$137$	196568.	0.894769	0.447385	+	0.894342i	$0.352355\pi$
$138$	−8681.40	−0.0388054
$139$	−29936.4	−0.131420	−0.0657102	−	0.997839i	$-0.520931\pi$
$139$	−29936.4	−0.131420	−0.0657102	+	0.997839i	$0.520931\pi$
$140$	0	0
$141$	176597.	0.748057
$142$	187859.	0.781830
$143$	−56120.0	−0.229497
$144$	146899.	0.590354
$145$	0	0
$146$	−242910.	−0.943111
$147$	241167.	0.920501
$148$	568.975	0.00213520
$149$	−362800.	−1.33876	−0.669379	−	0.742921i	$-0.733440\pi$
$149$	−362800.	−1.33876	−0.669379	+	0.742921i	$0.733440\pi$
$150$	0	0
$151$	226244.	0.807485	0.403742	−	0.914873i	$-0.367709\pi$
$151$	226244.	0.807485	0.403742	+	0.914873i	$0.367709\pi$
$152$	108386.	0.380507
$153$	−942182.	−3.25391
$154$	44276.7	0.150443
$155$	0	0
$156$	−212088.	−0.697758
$157$	−392007.	−1.26924	−0.634621	−	0.772824i	$-0.718844\pi$
$157$	−392007.	−1.26924	−0.634621	+	0.772824i	$0.718844\pi$
$158$	−73544.2	−0.234372
$159$	721794.	2.26423
$160$	0	0
$161$	−6946.97	−0.0211218
$162$	−523142.	−1.56615
$163$	84380.4	0.248755	0.124378	−	0.992235i	$-0.460307\pi$
$163$	84380.4	0.248755	0.124378	+	0.992235i	$0.460307\pi$
$164$	147556.	0.428398
$165$	0	0
$166$	−237288.	−0.668352
$167$	−543654.	−1.50845	−0.754226	−	0.656615i	$-0.771988\pi$
$167$	−543654.	−1.50845	−0.754226	+	0.656615i	$0.771988\pi$
$168$	167330.	0.457405
$169$	−156181.	−0.420642
$170$	0	0
$171$	−971785.	−2.54144
$172$	184267.	0.474925
$173$	−356699.	−0.906123	−0.453061	−	0.891479i	$-0.649668\pi$
$173$	−356699.	−0.906123	−0.453061	+	0.891479i	$0.649668\pi$
$174$	−336368.	−0.842252
$175$	0	0
$176$	−30976.0	−0.0753778
$177$	−1.16569e6	−2.79673
$178$	−20282.8	−0.0479820
$179$	−521697.	−1.21699	−0.608493	−	0.793559i	$-0.708226\pi$
$179$	−521697.	−1.21699	−0.608493	+	0.793559i	$0.708226\pi$
$180$	0	0
$181$	−698048.	−1.58376	−0.791879	−	0.610678i	$-0.790897\pi$
$181$	−698048.	−1.58376	−0.791879	+	0.610678i	$0.790897\pi$
$182$	−169716.	−0.379790
$183$	210603.	0.464875
$184$	4860.10	0.0105828
$185$	0	0
$186$	965280.	2.04584
$187$	198674.	0.415468
$188$	−98864.0	−0.204006
$189$	−864949.	−1.76131
$190$	0	0
$191$	612931.	1.21571	0.607853	−	0.794050i	$-0.292031\pi$
$191$	612931.	1.21571	0.607853	+	0.794050i	$0.292031\pi$
$192$	−117064.	−0.229177
$193$	184465.	0.356467	0.178234	−	0.983988i	$-0.442962\pi$
$193$	184465.	0.356467	0.178234	+	0.983988i	$0.442962\pi$
$194$	−523178.	−0.998034
$195$	0	0
$196$	−135012.	−0.251034
$197$	1.03211e6	1.89479	0.947393	−	0.320074i	$-0.103708\pi$
$197$	1.03211e6	1.89479	0.947393	+	0.320074i	$0.103708\pi$
$198$	277731.	0.503455
$199$	244463.	0.437603	0.218801	−	0.975769i	$-0.429785\pi$
$199$	244463.	0.437603	0.218801	+	0.975769i	$0.429785\pi$
$200$	0	0
$201$	−315088.	−0.550099
$202$	108459.	0.187020
$203$	−269166.	−0.458438
$204$	750828.	1.26318
$205$	0	0
$206$	−51211.6	−0.0840814
$207$	−43575.7	−0.0706835
$208$	118733.	0.190289
$209$	204917.	0.324498
$210$	0	0
$211$	1.12590e6	1.74098	0.870488	−	0.492190i	$-0.163803\pi$
$211$	1.12590e6	1.74098	0.870488	+	0.492190i	$0.163803\pi$
$212$	−404081.	−0.617489
$213$	1.34226e6	2.02716
$214$	−256526.	−0.382910
$215$	0	0
$216$	605119.	0.882483
$217$	772430.	1.11355
$218$	506519.	0.721863
$219$	−1.73560e6	−2.44534
$220$	0	0
$221$	−761532.	−1.04884
$222$	4065.35	0.00553624
$223$	−72016.8	−0.0969776	−0.0484888	−	0.998824i	$-0.515441\pi$
$223$	−72016.8	−0.0969776	−0.0484888	+	0.998824i	$0.515441\pi$
$224$	−93676.3	−0.124741
$225$	0	0
$226$	527813.	0.687400
$227$	−1.13240e6	−1.45860	−0.729298	−	0.684196i	$-0.760153\pi$
$227$	−1.13240e6	−1.45860	−0.729298	+	0.684196i	$0.760153\pi$
$228$	774419.	0.986595
$229$	−34423.7	−0.0433779	−0.0216890	−	0.999765i	$-0.506904\pi$
$229$	−34423.7	−0.0433779	−0.0216890	+	0.999765i	$0.506904\pi$
$230$	0	0
$231$	316358.	0.390076
$232$	188309.	0.229695
$233$	−689707.	−0.832290	−0.416145	−	0.909298i	$-0.636619\pi$
$233$	−689707.	−0.832290	−0.416145	+	0.909298i	$0.636619\pi$
$234$	−1.06456e6	−1.27096
$235$	0	0
$236$	652589.	0.762711
$237$	−525476.	−0.607690
$238$	600822.	0.687549
$239$	289978.	0.328375	0.164188	−	0.986429i	$-0.447500\pi$
$239$	289978.	0.328375	0.164188	+	0.986429i	$0.447500\pi$
$240$	0	0
$241$	1.40673e6	1.56016	0.780078	−	0.625682i	$-0.215179\pi$
$241$	1.40673e6	1.56016	0.780078	+	0.625682i	$0.215179\pi$
$242$	−58564.0	−0.0642824
$243$	−1.44030e6	−1.56473
$244$	−117902.	−0.126778
$245$	0	0
$246$	1.05429e6	1.11077
$247$	−785459.	−0.819184
$248$	−540392.	−0.557930
$249$	−1.69543e6	−1.73293
$250$	0	0
$251$	580131.	0.581222	0.290611	−	0.956841i	$-0.406141\pi$
$251$	580131.	0.581222	0.290611	+	0.956841i	$0.406141\pi$
$252$	839901.	0.833157
$253$	9188.63	0.00902505
$254$	475505.	0.462457
$255$	0	0
$256$	65536.0	0.0625000
$257$	407104.	0.384479	0.192240	−	0.981348i	$-0.438425\pi$
$257$	407104.	0.384479	0.192240	+	0.981348i	$0.438425\pi$
$258$	1.31659e6	1.23141
$259$	3253.14	0.00301338
$260$	0	0
$261$	−1.68838e6	−1.53415
$262$	−823739.	−0.741372
$263$	1.57675e6	1.40564	0.702819	−	0.711369i	$-0.251924\pi$
$263$	1.57675e6	1.40564	0.702819	+	0.711369i	$0.251924\pi$
$264$	−221325.	−0.195443
$265$	0	0
$266$	619700.	0.537004
$267$	−144921.	−0.124410
$268$	176395.	0.150020
$269$	−752593.	−0.634132	−0.317066	−	0.948404i	$-0.602698\pi$
$269$	−752593.	−0.634132	−0.317066	+	0.948404i	$0.602698\pi$
$270$	0	0
$271$	−208553.	−0.172501	−0.0862507	−	0.996273i	$-0.527489\pi$
$271$	−208553.	−0.172501	−0.0862507	+	0.996273i	$0.527489\pi$
$272$	−420336.	−0.344488
$273$	−1.21262e6	−0.984735
$274$	−786272.	−0.632698
$275$	0	0
$276$	34725.6	0.0274395
$277$	−1.01926e6	−0.798151	−0.399075	−	0.916918i	$-0.630669\pi$
$277$	−1.01926e6	−0.798151	−0.399075	+	0.916918i	$0.630669\pi$
$278$	119746.	0.0929282
$279$	4.84516e6	3.72647
$280$	0	0
$281$	865597.	0.653958	0.326979	−	0.945032i	$-0.393969\pi$
$281$	865597.	0.653958	0.326979	+	0.945032i	$0.393969\pi$
$282$	−706386.	−0.528956
$283$	−1.72508e6	−1.28040	−0.640198	−	0.768210i	$-0.721148\pi$
$283$	−1.72508e6	−1.28040	−0.640198	+	0.768210i	$0.721148\pi$
$284$	−751438.	−0.552837
$285$	0	0
$286$	224480.	0.162279
$287$	843658.	0.604591
$288$	−587595.	−0.417443
$289$	1.27610e6	0.898750
$290$	0	0
$291$	−3.73813e6	−2.58774
$292$	971639.	0.666880
$293$	−1.25254e6	−0.852361	−0.426181	−	0.904638i	$-0.640141\pi$
$293$	−1.25254e6	−0.852361	−0.426181	+	0.904638i	$0.640141\pi$
$294$	−964667.	−0.650892
$295$	0	0
$296$	−2275.90	−0.00150982
$297$	1.14405e6	0.752584
$298$	1.45120e6	0.946645
$299$	−35220.7	−0.0227834
$300$	0	0
$301$	1.05355e6	0.670255
$302$	−904975.	−0.570978
$303$	774945.	0.484914
$304$	−433543.	−0.269059
$305$	0	0
$306$	3.76873e6	2.30086
$307$	−2.69819e6	−1.63390	−0.816951	−	0.576707i	$-0.804337\pi$
$307$	−2.69819e6	−1.63390	−0.816951	+	0.576707i	$0.804337\pi$
$308$	−177107.	−0.106380
$309$	−365908.	−0.218010
$310$	0	0
$311$	−1.74373e6	−1.02230	−0.511150	−	0.859491i	$-0.670780\pi$
$311$	−1.74373e6	−1.02230	−0.511150	+	0.859491i	$0.670780\pi$
$312$	848352.	0.493389
$313$	1.15038e6	0.663714	0.331857	−	0.943330i	$-0.392325\pi$
$313$	1.15038e6	0.663714	0.331857	+	0.943330i	$0.392325\pi$
$314$	1.56803e6	0.897489
$315$	0	0
$316$	294177.	0.165726
$317$	1.16110e6	0.648967	0.324484	−	0.945891i	$-0.394810\pi$
$317$	1.16110e6	0.648967	0.324484	+	0.945891i	$0.394810\pi$
$318$	−2.88717e6	−1.60105
$319$	356022.	0.195884
$320$	0	0
$321$	−1.83289e6	−0.992825
$322$	27787.9	0.0149354
$323$	2.78066e6	1.48300
$324$	2.09257e6	1.10743
$325$	0	0
$326$	−337522.	−0.175897
$327$	3.61910e6	1.87168
$328$	−590224.	−0.302923
$329$	−565260.	−0.287911
$330$	0	0
$331$	−2.31915e6	−1.16348	−0.581740	−	0.813375i	$-0.697628\pi$
$331$	−2.31915e6	−1.16348	−0.581740	+	0.813375i	$0.697628\pi$
$332$	949151.	0.472596
$333$	20405.7	0.0100842
$334$	2.17462e6	1.06664
$335$	0	0
$336$	−669320.	−0.323434
$337$	−1.43786e6	−0.689671	−0.344835	−	0.938663i	$-0.612065\pi$
$337$	−1.43786e6	−0.689671	−0.344835	+	0.938663i	$0.612065\pi$
$338$	624725.	0.297439
$339$	3.77124e6	1.78232
$340$	0	0
$341$	−1.02168e6	−0.475805
$342$	3.88714e6	1.79707
$343$	−2.30946e6	−1.05992
$344$	−737066.	−0.335823
$345$	0	0
$346$	1.42680e6	0.640726
$347$	−575629.	−0.256637	−0.128318	−	0.991733i	$-0.540958\pi$
$347$	−575629.	−0.256637	−0.128318	+	0.991733i	$0.540958\pi$
$348$	1.34547e6	0.595562
$349$	−2.77702e6	−1.22044	−0.610219	−	0.792233i	$-0.708919\pi$
$349$	−2.77702e6	−1.22044	−0.610219	+	0.792233i	$0.708919\pi$
$350$	0	0
$351$	−4.38523e6	−1.89987
$352$	123904.	0.0533002
$353$	−2.18172e6	−0.931885	−0.465942	−	0.884815i	$-0.654285\pi$
$353$	−2.18172e6	−0.931885	−0.465942	+	0.884815i	$0.654285\pi$
$354$	4.66277e6	1.97759
$355$	0	0
$356$	81131.3	0.0339284
$357$	4.29289e6	1.78271
$358$	2.08679e6	0.860539
$359$	−2.14848e6	−0.879822	−0.439911	−	0.898041i	$-0.644990\pi$
$359$	−2.14848e6	−0.879822	−0.439911	+	0.898041i	$0.644990\pi$
$360$	0	0
$361$	391930.	0.158285
$362$	2.79219e6	1.11989
$363$	−418442.	−0.166674
$364$	678862.	0.268552
$365$	0	0
$366$	−842411.	−0.328716
$367$	−15838.0	−0.00613813	−0.00306907	−	0.999995i	$-0.500977\pi$
$367$	−15838.0	−0.00613813	−0.00306907	+	0.999995i	$0.500977\pi$
$368$	−19440.4	−0.00748317
$369$	5.29195e6	2.02325
$370$	0	0
$371$	−2.31035e6	−0.871453
$372$	−3.86112e6	−1.44663
$373$	394343.	0.146758	0.0733791	−	0.997304i	$-0.476622\pi$
$373$	394343.	0.146758	0.0733791	+	0.997304i	$0.476622\pi$
$374$	−794697.	−0.293780
$375$	0	0
$376$	395456.	0.144254
$377$	−1.36465e6	−0.494504
$378$	3.45980e6	1.24544
$379$	1.68501e6	0.602567	0.301283	−	0.953535i	$-0.402585\pi$
$379$	1.68501e6	0.602567	0.301283	+	0.953535i	$0.402585\pi$
$380$	0	0
$381$	3.39750e6	1.19908
$382$	−2.45172e6	−0.859633
$383$	−3.94284e6	−1.37345	−0.686724	−	0.726918i	$-0.740952\pi$
$383$	−3.94284e6	−1.37345	−0.686724	+	0.726918i	$0.740952\pi$
$384$	468257.	0.162053
$385$	0	0
$386$	−737859.	−0.252061
$387$	6.60853e6	2.24299
$388$	2.09271e6	0.705717
$389$	−571118.	−0.191360	−0.0956801	−	0.995412i	$-0.530503\pi$
$389$	−571118.	−0.191360	−0.0956801	+	0.995412i	$0.530503\pi$
$390$	0	0
$391$	124687.	0.0412458
$392$	540049.	0.177508
$393$	−5.88564e6	−1.92226
$394$	−4.12844e6	−1.33982
$395$	0	0
$396$	−1.11092e6	−0.355997
$397$	−6.17025e6	−1.96484	−0.982419	−	0.186689i	$-0.940224\pi$
$397$	−6.17025e6	−1.96484	−0.982419	+	0.186689i	$0.940224\pi$
$398$	−977851.	−0.309432
$399$	4.42778e6	1.39237
$400$	0	0
$401$	1.82838e6	0.567814	0.283907	−	0.958852i	$-0.408369\pi$
$401$	1.82838e6	0.567814	0.283907	+	0.958852i	$0.408369\pi$
$402$	1.26035e6	0.388979
$403$	3.91617e6	1.20115
$404$	−433837.	−0.132243
$405$	0	0
$406$	1.07667e6	0.324165
$407$	−4302.88	−0.00128758
$408$	−3.00331e6	−0.893202
$409$	−5.94235e6	−1.75651	−0.878254	−	0.478195i	$-0.841291\pi$
$409$	−5.94235e6	−1.75651	−0.878254	+	0.478195i	$0.841291\pi$
$410$	0	0
$411$	−5.61794e6	−1.64048
$412$	204846.	0.0594546
$413$	3.73121e6	1.07640
$414$	174303.	0.0499808
$415$	0	0
$416$	−474933.	−0.134555
$417$	855586.	0.240948
$418$	−819666.	−0.229454
$419$	−2.80473e6	−0.780468	−0.390234	−	0.920716i	$-0.627606\pi$
$419$	−2.80473e6	−0.780468	−0.390234	+	0.920716i	$0.627606\pi$
$420$	0	0
$421$	3.57571e6	0.983234	0.491617	−	0.870812i	$-0.336406\pi$
$421$	3.57571e6	0.983234	0.491617	+	0.870812i	$0.336406\pi$
$422$	−4.50359e6	−1.23106
$423$	−3.54566e6	−0.963487
$424$	1.61633e6	0.436631
$425$	0	0
$426$	−5.36905e6	−1.43342
$427$	−674108.	−0.178920
$428$	1.02610e6	0.270758
$429$	1.60392e6	0.420764
$430$	0	0
$431$	−2.82711e6	−0.733078	−0.366539	−	0.930403i	$-0.619457\pi$
$431$	−2.82711e6	−0.733078	−0.366539	+	0.930403i	$0.619457\pi$
$432$	−2.42048e6	−0.624010
$433$	765652.	0.196251	0.0981254	−	0.995174i	$-0.468715\pi$
$433$	765652.	0.196251	0.0981254	+	0.995174i	$0.468715\pi$
$434$	−3.08972e6	−0.787399
$435$	0	0
$436$	−2.02608e6	−0.510434
$437$	128605.	0.0322147
$438$	6.94239e6	1.72912
$439$	4.65788e6	1.15352	0.576762	−	0.816912i	$-0.304316\pi$
$439$	4.65788e6	1.15352	0.576762	+	0.816912i	$0.304316\pi$
$440$	0	0
$441$	−4.84208e6	−1.18559
$442$	3.04613e6	0.741639
$443$	4.99700e6	1.20976	0.604881	−	0.796316i	$-0.293221\pi$
$443$	4.99700e6	1.20976	0.604881	+	0.796316i	$0.293221\pi$
$444$	−16261.4	−0.00391471
$445$	0	0
$446$	288067.	0.0685735
$447$	1.03689e7	2.45450
$448$	374705.	0.0882053
$449$	165769.	0.0388050	0.0194025	−	0.999812i	$-0.493824\pi$
$449$	165769.	0.0388050	0.0194025	+	0.999812i	$0.493824\pi$
$450$	0	0
$451$	−1.11589e6	−0.258334
$452$	−2.11125e6	−0.486065
$453$	−6.46608e6	−1.48046
$454$	4.52960e6	1.03138
$455$	0	0
$456$	−3.09768e6	−0.697628
$457$	8.07259e6	1.80810	0.904050	−	0.427428i	$-0.140580\pi$
$457$	8.07259e6	1.80810	0.904050	+	0.427428i	$0.140580\pi$
$458$	137695.	0.0306728
$459$	1.55245e7	3.43942
$460$	0	0
$461$	−5.87512e6	−1.28755	−0.643776	−	0.765214i	$-0.722633\pi$
$461$	−5.87512e6	−1.28755	−0.643776	+	0.765214i	$0.722633\pi$
$462$	−1.26543e6	−0.275825
$463$	1.02289e6	0.221757	0.110878	−	0.993834i	$-0.464634\pi$
$463$	1.02289e6	0.221757	0.110878	+	0.993834i	$0.464634\pi$
$464$	−753236.	−0.162419
$465$	0	0
$466$	2.75883e6	0.588518
$467$	6.17201e6	1.30959	0.654794	−	0.755807i	$-0.272755\pi$
$467$	6.17201e6	1.30959	0.654794	+	0.755807i	$0.272755\pi$
$468$	4.25824e6	0.898702
$469$	1.00855e6	0.211721
$470$	0	0
$471$	1.12036e7	2.32705
$472$	−2.61036e6	−0.539318
$473$	−1.39352e6	−0.286391
$474$	2.10190e6	0.429702
$475$	0	0
$476$	−2.40329e6	−0.486170
$477$	−1.44920e7	−2.91629
$478$	−1.15991e6	−0.232196
$479$	−7.00244e6	−1.39447	−0.697237	−	0.716841i	$-0.745588\pi$
$479$	−7.00244e6	−1.39447	−0.697237	+	0.716841i	$0.745588\pi$
$480$	0	0
$481$	16493.2	0.00325044
$482$	−5.62692e6	−1.10320
$483$	198545.	0.0387250
$484$	234256.	0.0454545
$485$	0	0
$486$	5.76122e6	1.10643
$487$	−365769.	−0.0698852	−0.0349426	−	0.999389i	$-0.511125\pi$
$487$	−365769.	−0.0698852	−0.0349426	+	0.999389i	$0.511125\pi$
$488$	471607.	0.0896459
$489$	−2.41160e6	−0.456072
$490$	0	0
$491$	5.00170e6	0.936297	0.468149	−	0.883650i	$-0.344921\pi$
$491$	5.00170e6	0.936297	0.468149	+	0.883650i	$0.344921\pi$
$492$	−4.21717e6	−0.785431
$493$	4.83111e6	0.895220
$494$	3.14184e6	0.579250
$495$	0	0
$496$	2.16157e6	0.394516
$497$	−4.29638e6	−0.780211
$498$	6.78171e6	1.22537
$499$	−1.63517e6	−0.293976	−0.146988	−	0.989138i	$-0.546958\pi$
$499$	−1.63517e6	−0.293976	−0.146988	+	0.989138i	$0.546958\pi$
$500$	0	0
$501$	1.55377e7	2.76562
$502$	−2.32053e6	−0.410986
$503$	−5.20393e6	−0.917088	−0.458544	−	0.888672i	$-0.651629\pi$
$503$	−5.20393e6	−0.917088	−0.458544	+	0.888672i	$0.651629\pi$
$504$	−3.35961e6	−0.589131
$505$	0	0
$506$	−36754.5	−0.00638167
$507$	4.46368e6	0.771212
$508$	−1.90202e6	−0.327006
$509$	−8.62390e6	−1.47540	−0.737699	−	0.675130i	$-0.764088\pi$
$509$	−8.62390e6	−1.47540	−0.737699	+	0.675130i	$0.764088\pi$
$510$	0	0
$511$	5.55539e6	0.941158
$512$	−262144.	−0.0441942
$513$	1.60123e7	2.68633
$514$	−1.62842e6	−0.271868
$515$	0	0
$516$	−5.26636e6	−0.870736
$517$	747659.	0.123020
$518$	−13012.6	−0.00213078
$519$	1.01945e7	1.66130
$520$	0	0
$521$	140764.	0.0227194	0.0113597	−	0.999935i	$-0.496384\pi$
$521$	140764.	0.0227194	0.0113597	+	0.999935i	$0.496384\pi$
$522$	6.75351e6	1.08481
$523$	9.29900e6	1.48656	0.743279	−	0.668981i	$-0.233269\pi$
$523$	9.29900e6	1.48656	0.743279	+	0.668981i	$0.233269\pi$
$524$	3.29496e6	0.524229
$525$	0	0
$526$	−6.30700e6	−0.993936
$527$	−1.38639e7	−2.17450
$528$	885298.	0.138199
$529$	−6.43058e6	−0.999104
$530$	0	0
$531$	2.34044e7	3.60215
$532$	−2.47880e6	−0.379719
$533$	4.27729e6	0.652155
$534$	579686.	0.0879710
$535$	0	0
$536$	−705581.	−0.106080
$537$	1.49102e7	2.23124
$538$	3.01037e6	0.448399
$539$	1.02103e6	0.151379
$540$	0	0
$541$	−4.98590e6	−0.732403	−0.366202	−	0.930536i	$-0.619342\pi$
$541$	−4.98590e6	−0.732403	−0.366202	+	0.930536i	$0.619342\pi$
$542$	834211.	0.121977
$543$	1.99503e7	2.90369
$544$	1.68134e6	0.243590
$545$	0	0
$546$	4.85049e6	0.696313
$547$	3.57048e6	0.510221	0.255111	−	0.966912i	$-0.417888\pi$
$547$	3.57048e6	0.510221	0.255111	+	0.966912i	$0.417888\pi$
$548$	3.14509e6	0.447385
$549$	−4.22842e6	−0.598753
$550$	0	0
$551$	4.98291e6	0.699204
$552$	−138902.	−0.0194027
$553$	1.68197e6	0.233887
$554$	4.07703e6	0.564378
$555$	0	0
$556$	−478983.	−0.0657102
$557$	−1.24799e7	−1.70441	−0.852206	−	0.523207i	$-0.824736\pi$
$557$	−1.24799e7	−1.70441	−0.852206	+	0.523207i	$0.824736\pi$
$558$	−1.93806e7	−2.63501
$559$	5.34144e6	0.722984
$560$	0	0
$561$	−5.67814e6	−0.761726
$562$	−3.46239e6	−0.462418
$563$	5.12581e6	0.681540	0.340770	−	0.940147i	$-0.389312\pi$
$563$	5.12581e6	0.681540	0.340770	+	0.940147i	$0.389312\pi$
$564$	2.82555e6	0.374028
$565$	0	0
$566$	6.90033e6	0.905376
$567$	1.19644e7	1.56290
$568$	3.00575e6	0.390915
$569$	−1.09610e6	−0.141929	−0.0709645	−	0.997479i	$-0.522608\pi$
$569$	−1.09610e6	−0.141929	−0.0709645	+	0.997479i	$0.522608\pi$
$570$	0	0
$571$	−1.08787e7	−1.39632	−0.698162	−	0.715940i	$-0.745998\pi$
$571$	−1.08787e7	−1.39632	−0.698162	+	0.715940i	$0.745998\pi$
$572$	−897919.	−0.114749
$573$	−1.75176e7	−2.22889
$574$	−3.37463e6	−0.427511
$575$	0	0
$576$	2.35038e6	0.295177
$577$	−1.37408e7	−1.71820	−0.859101	−	0.511807i	$-0.828976\pi$
$577$	−1.37408e7	−1.71820	−0.859101	+	0.511807i	$0.828976\pi$
$578$	−5.10439e6	−0.635512
$579$	−5.27202e6	−0.653553
$580$	0	0
$581$	5.42681e6	0.666967
$582$	1.49525e7	1.82981
$583$	3.05587e6	0.372360
$584$	−3.88656e6	−0.471556
$585$	0	0
$586$	5.01017e6	0.602711
$587$	−7.50167e6	−0.898593	−0.449296	−	0.893383i	$-0.648325\pi$
$587$	−7.50167e6	−0.898593	−0.449296	+	0.893383i	$0.648325\pi$
$588$	3.85867e6	0.460250
$589$	−1.42995e7	−1.69837
$590$	0	0
$591$	−2.94978e7	−3.47393
$592$	9103.60	0.00106760
$593$	−1.23281e7	−1.43966	−0.719829	−	0.694151i	$-0.755780\pi$
$593$	−1.23281e7	−1.43966	−0.719829	+	0.694151i	$0.755780\pi$
$594$	−4.57621e6	−0.532157
$595$	0	0
$596$	−5.80481e6	−0.669379
$597$	−6.98677e6	−0.802308
$598$	140883.	0.0161103
$599$	3.71959e6	0.423573	0.211786	−	0.977316i	$-0.432072\pi$
$599$	3.71959e6	0.423573	0.211786	+	0.977316i	$0.432072\pi$
$600$	0	0
$601$	−8.89207e6	−1.00419	−0.502096	−	0.864812i	$-0.667437\pi$
$601$	−8.89207e6	−1.00419	−0.502096	+	0.864812i	$0.667437\pi$
$602$	−4.21421e6	−0.473942
$603$	6.32624e6	0.708520
$604$	3.61990e6	0.403742
$605$	0	0
$606$	−3.09978e6	−0.342886
$607$	−1.18740e7	−1.30805	−0.654027	−	0.756471i	$-0.726922\pi$
$607$	−1.18740e7	−1.30805	−0.654027	+	0.756471i	$0.726922\pi$
$608$	1.73417e6	0.190254
$609$	7.69281e6	0.840508
$610$	0	0
$611$	−2.86583e6	−0.310561
$612$	−1.50749e7	−1.62696
$613$	−9.08062e6	−0.976033	−0.488016	−	0.872834i	$-0.662279\pi$
$613$	−9.08062e6	−0.976033	−0.488016	+	0.872834i	$0.662279\pi$
$614$	1.07927e7	1.15534
$615$	0	0
$616$	708427.	0.0752217
$617$	9.98190e6	1.05560	0.527801	−	0.849368i	$-0.323017\pi$
$617$	9.98190e6	1.05560	0.527801	+	0.849368i	$0.323017\pi$
$618$	1.46363e6	0.154156
$619$	−3.86586e6	−0.405527	−0.202763	−	0.979228i	$-0.564992\pi$
$619$	−3.86586e6	−0.405527	−0.202763	+	0.979228i	$0.564992\pi$
$620$	0	0
$621$	718003.	0.0747132
$622$	6.97492e6	0.722876
$623$	463872.	0.0478827
$624$	−3.39341e6	−0.348879
$625$	0	0
$626$	−4.60152e6	−0.469316
$627$	−5.85654e6	−0.594939
$628$	−6.27211e6	−0.634621
$629$	−58388.8	−0.00588441
$630$	0	0
$631$	1.45626e7	1.45601	0.728007	−	0.685570i	$-0.240447\pi$
$631$	1.45626e7	1.45601	0.728007	+	0.685570i	$0.240447\pi$
$632$	−1.17671e6	−0.117186
$633$	−3.21783e7	−3.19193
$634$	−4.64441e6	−0.458889
$635$	0	0
$636$	1.15487e7	1.13211
$637$	−3.91368e6	−0.382153
$638$	−1.42409e6	−0.138511
$639$	−2.69496e7	−2.61096
$640$	0	0
$641$	−1.34201e7	−1.29006	−0.645032	−	0.764155i	$-0.723156\pi$
$641$	−1.34201e7	−1.29006	−0.645032	+	0.764155i	$0.723156\pi$
$642$	7.33154e6	0.702033
$643$	−1.50895e7	−1.43929	−0.719644	−	0.694344i	$-0.755695\pi$
$643$	−1.50895e7	−1.43929	−0.719644	+	0.694344i	$0.755695\pi$
$644$	−111151.	−0.0105609
$645$	0	0
$646$	−1.11226e7	−1.04864
$647$	−1.42974e7	−1.34275	−0.671377	−	0.741116i	$-0.734297\pi$
$647$	−1.42974e7	−1.34275	−0.671377	+	0.741116i	$0.734297\pi$
$648$	−8.37027e6	−0.783073
$649$	−4.93520e6	−0.459932
$650$	0	0
$651$	−2.20761e7	−2.04160
$652$	1.35009e6	0.124378
$653$	1.31607e6	0.120780	0.0603900	−	0.998175i	$-0.480766\pi$
$653$	1.31607e6	0.120780	0.0603900	+	0.998175i	$0.480766\pi$
$654$	−1.44764e7	−1.32348
$655$	0	0
$656$	2.36090e6	0.214199
$657$	3.48469e7	3.14956
$658$	2.26104e6	0.203584
$659$	3.75917e6	0.337193	0.168596	−	0.985685i	$-0.446077\pi$
$659$	3.75917e6	0.337193	0.168596	+	0.985685i	$0.446077\pi$
$660$	0	0
$661$	−8.95170e6	−0.796896	−0.398448	−	0.917191i	$-0.630451\pi$
$661$	−8.95170e6	−0.796896	−0.398448	+	0.917191i	$0.630451\pi$
$662$	9.27660e6	0.822705
$663$	2.17647e7	1.92295
$664$	−3.79660e6	−0.334176
$665$	0	0
$666$	−81622.9	−0.00713060
$667$	223438.	0.0194465
$668$	−8.69847e6	−0.754226
$669$	2.05825e6	0.177800
$670$	0	0
$671$	891631.	0.0764503
$672$	2.67728e6	0.228702
$673$	519050.	0.0441745	0.0220873	−	0.999756i	$-0.492969\pi$
$673$	519050.	0.0441745	0.0220873	+	0.999756i	$0.492969\pi$
$674$	5.75144e6	0.487671
$675$	0	0
$676$	−2.49890e6	−0.210321
$677$	4.12272e6	0.345711	0.172855	−	0.984947i	$-0.444701\pi$
$677$	4.12272e6	0.345711	0.172855	+	0.984947i	$0.444701\pi$
$678$	−1.50850e7	−1.26029
$679$	1.19652e7	0.995967
$680$	0	0
$681$	3.23641e7	2.67421
$682$	4.08672e6	0.336445
$683$	−8.35625e6	−0.685425	−0.342712	−	0.939440i	$-0.611346\pi$
$683$	−8.35625e6	−0.685425	−0.342712	+	0.939440i	$0.611346\pi$
$684$	−1.55486e7	−1.27072
$685$	0	0
$686$	9.23783e6	0.749479
$687$	983834.	0.0795298
$688$	2.94826e6	0.237463
$689$	−1.17133e7	−0.940011
$690$	0	0
$691$	9.40353e6	0.749197	0.374598	−	0.927187i	$-0.377781\pi$
$691$	9.40353e6	0.749197	0.374598	+	0.927187i	$0.377781\pi$
$692$	−5.70719e6	−0.453061
$693$	−6.35175e6	−0.502413
$694$	2.30252e6	0.181470
$695$	0	0
$696$	−5.38189e6	−0.421126
$697$	−1.51423e7	−1.18062
$698$	1.11081e7	0.862980
$699$	1.97119e7	1.52593
$700$	0	0
$701$	−1.76888e7	−1.35957	−0.679787	−	0.733410i	$-0.737928\pi$
$701$	−1.76888e7	−1.35957	−0.679787	+	0.733410i	$0.737928\pi$
$702$	1.75409e7	1.34341
$703$	−60223.4	−0.00459597
$704$	−495616.	−0.0376889
$705$	0	0
$706$	8.72688e6	0.658942
$707$	−2.48049e6	−0.186633
$708$	−1.86511e7	−1.39837
$709$	−1.11343e7	−0.831853	−0.415926	−	0.909398i	$-0.636543\pi$
$709$	−1.11343e7	−0.831853	−0.415926	+	0.909398i	$0.636543\pi$
$710$	0	0
$711$	1.05504e7	0.782696
$712$	−324525.	−0.0239910
$713$	−641202.	−0.0472358
$714$	−1.71716e7	−1.26056
$715$	0	0
$716$	−8.34715e6	−0.608493
$717$	−8.28761e6	−0.602048
$718$	8.59391e6	0.622128
$719$	−9.55010e6	−0.688947	−0.344474	−	0.938796i	$-0.611943\pi$
$719$	−9.55010e6	−0.688947	−0.344474	+	0.938796i	$0.611943\pi$
$720$	0	0
$721$	1.17122e6	0.0839073
$722$	−1.56772e6	−0.111925
$723$	−4.02045e7	−2.86042
$724$	−1.11688e7	−0.791879
$725$	0	0
$726$	1.67377e6	0.117856
$727$	1.89994e7	1.33322	0.666612	−	0.745405i	$-0.267744\pi$
$727$	1.89994e7	1.33322	0.666612	+	0.745405i	$0.267744\pi$
$728$	−2.71545e6	−0.189895
$729$	9.38322e6	0.653933
$730$	0	0
$731$	−1.89096e7	−1.30885
$732$	3.36964e6	0.232438
$733$	2.68111e7	1.84312	0.921561	−	0.388233i	$-0.126914\pi$
$733$	2.68111e7	1.84312	0.921561	+	0.388233i	$0.126914\pi$
$734$	63352.2	0.00434032
$735$	0	0
$736$	77761.6	0.00529140
$737$	−1.33399e6	−0.0904657
$738$	−2.11678e7	−1.43065
$739$	−6.41899e6	−0.432370	−0.216185	−	0.976352i	$-0.569361\pi$
$739$	−6.41899e6	−0.432370	−0.216185	+	0.976352i	$0.569361\pi$
$740$	0	0
$741$	2.24485e7	1.50190
$742$	9.24142e6	0.616210
$743$	−4.66177e6	−0.309798	−0.154899	−	0.987930i	$-0.549505\pi$
$743$	−4.66177e6	−0.309798	−0.154899	+	0.987930i	$0.549505\pi$
$744$	1.54445e7	1.02292
$745$	0	0
$746$	−1.57737e6	−0.103774
$747$	3.40403e7	2.23199
$748$	3.17879e6	0.207734
$749$	5.86679e6	0.382117
$750$	0	0
$751$	2.39792e7	1.55144	0.775721	−	0.631076i	$-0.217386\pi$
$751$	2.39792e7	1.55144	0.775721	+	0.631076i	$0.217386\pi$
$752$	−1.58182e6	−0.102003
$753$	−1.65802e7	−1.06562
$754$	5.45862e6	0.349667
$755$	0	0
$756$	−1.38392e7	−0.880656
$757$	−4.65197e6	−0.295051	−0.147526	−	0.989058i	$-0.547131\pi$
$757$	−4.65197e6	−0.295051	−0.147526	+	0.989058i	$0.547131\pi$
$758$	−6.74005e6	−0.426079
$759$	−262612.	−0.0165467
$760$	0	0
$761$	−1.91952e7	−1.20152	−0.600760	−	0.799429i	$-0.705135\pi$
$761$	−1.91952e7	−1.20152	−0.600760	+	0.799429i	$0.705135\pi$
$762$	−1.35900e7	−0.847876
$763$	−1.15842e7	−0.720368
$764$	9.80690e6	0.607853
$765$	0	0
$766$	1.57714e7	0.971175
$767$	1.89170e7	1.16108
$768$	−1.87303e6	−0.114589
$769$	1.47548e7	0.899743	0.449871	−	0.893093i	$-0.351470\pi$
$769$	1.47548e7	0.899743	0.449871	+	0.893093i	$0.351470\pi$
$770$	0	0
$771$	−1.16351e7	−0.704910
$772$	2.95143e6	0.178234
$773$	−1.48823e7	−0.895820	−0.447910	−	0.894079i	$-0.647831\pi$
$773$	−1.48823e7	−0.895820	−0.447910	+	0.894079i	$0.647831\pi$
$774$	−2.64341e7	−1.58603
$775$	0	0
$776$	−8.37085e6	−0.499017
$777$	−92975.2	−0.00552478
$778$	2.28447e6	0.135312
$779$	−1.56181e7	−0.922115
$780$	0	0
$781$	5.68275e6	0.333373
$782$	−498749.	−0.0291652
$783$	2.78197e7	1.62161
$784$	−2.16020e6	−0.125517
$785$	0	0
$786$	2.35426e7	1.35924
$787$	−1.56041e7	−0.898054	−0.449027	−	0.893518i	$-0.648229\pi$
$787$	−1.56041e7	−0.898054	−0.449027	+	0.893518i	$0.648229\pi$
$788$	1.65137e7	0.947393
$789$	−4.50637e7	−2.57712
$790$	0	0
$791$	−1.20712e7	−0.685976
$792$	4.44369e6	0.251728
$793$	−3.41768e6	−0.192996
$794$	2.46810e7	1.38935
$795$	0	0
$796$	3.91140e6	0.218801
$797$	−2.19852e7	−1.22599	−0.612993	−	0.790088i	$-0.710035\pi$
$797$	−2.19852e7	−1.22599	−0.612993	+	0.790088i	$0.710035\pi$
$798$	−1.77111e7	−0.984552
$799$	1.01455e7	0.562221
$800$	0	0
$801$	2.90969e6	0.160238
$802$	−7.31353e6	−0.401505
$803$	−7.34802e6	−0.402144
$804$	−5.04140e6	−0.275050
$805$	0	0
$806$	−1.56647e7	−0.849344
$807$	2.15092e7	1.16263
$808$	1.73535e6	0.0935101
$809$	2.91738e7	1.56719	0.783596	−	0.621271i	$-0.213383\pi$
$809$	2.91738e7	1.56719	0.783596	+	0.621271i	$0.213383\pi$
$810$	0	0
$811$	2.59750e7	1.38677	0.693383	−	0.720569i	$-0.256119\pi$
$811$	2.59750e7	1.38677	0.693383	+	0.720569i	$0.256119\pi$
$812$	−4.30666e6	−0.229219
$813$	5.96046e6	0.316267
$814$	17211.5	0.000910453 0
$815$	0	0
$816$	1.20132e7	0.631590
$817$	−1.95038e7	−1.02226
$818$	2.37694e7	1.24204
$819$	2.43467e7	1.26832
$820$	0	0
$821$	3.42686e7	1.77435	0.887174	−	0.461436i	$-0.152666\pi$
$821$	3.42686e7	1.77435	0.887174	+	0.461436i	$0.152666\pi$
$822$	2.24717e7	1.16000
$823$	2.37132e7	1.22037	0.610184	−	0.792259i	$-0.291095\pi$
$823$	2.37132e7	1.22037	0.610184	+	0.792259i	$0.291095\pi$
$824$	−819385.	−0.0420407
$825$	0	0
$826$	−1.49248e7	−0.761131
$827$	−1.83274e7	−0.931833	−0.465916	−	0.884829i	$-0.654275\pi$
$827$	−1.83274e7	−0.931833	−0.465916	+	0.884829i	$0.654275\pi$
$828$	−697211.	−0.0353418
$829$	−2.66383e7	−1.34623	−0.673115	−	0.739538i	$-0.735044\pi$
$829$	−2.66383e7	−1.34623	−0.673115	+	0.739538i	$0.735044\pi$
$830$	0	0
$831$	2.91305e7	1.46334
$832$	1.89973e6	0.0951445
$833$	1.38551e7	0.691826
$834$	−3.42235e6	−0.170376
$835$	0	0
$836$	3.27867e6	0.162249
$837$	−7.98344e7	−3.93891
$838$	1.12189e7	0.551874
$839$	1.46709e7	0.719537	0.359768	−	0.933042i	$-0.382856\pi$
$839$	1.46709e7	0.719537	0.359768	+	0.933042i	$0.382856\pi$
$840$	0	0
$841$	−1.18539e7	−0.577923
$842$	−1.43028e7	−0.695251
$843$	−2.47389e7	−1.19898
$844$	1.80144e7	0.870488
$845$	0	0
$846$	1.41826e7	0.681288
$847$	1.33937e6	0.0641493
$848$	−6.46530e6	−0.308745
$849$	4.93031e7	2.34750
$850$	0	0
$851$	−2700.47	−0.000127825 0
$852$	2.14762e7	1.01358
$853$	1.74709e7	0.822133	0.411066	−	0.911605i	$-0.365156\pi$
$853$	1.74709e7	0.822133	0.411066	+	0.911605i	$0.365156\pi$
$854$	2.69643e6	0.126516
$855$	0	0
$856$	−4.10441e6	−0.191455
$857$	2.49580e7	1.16080	0.580402	−	0.814330i	$-0.302896\pi$
$857$	2.49580e7	1.16080	0.580402	+	0.814330i	$0.302896\pi$
$858$	−6.41566e6	−0.297525
$859$	−1.42612e7	−0.659439	−0.329719	−	0.944079i	$-0.606954\pi$
$859$	−1.42612e7	−0.659439	−0.329719	+	0.944079i	$0.606954\pi$
$860$	0	0
$861$	−2.41119e7	−1.10847
$862$	1.13085e7	0.518364
$863$	−494514.	−0.0226022	−0.0113011	−	0.999936i	$-0.503597\pi$
$863$	−494514.	−0.0226022	−0.0113011	+	0.999936i	$0.503597\pi$
$864$	9.68190e6	0.441242
$865$	0	0
$866$	−3.06261e6	−0.138770
$867$	−3.64710e7	−1.64778
$868$	1.23589e7	0.556775
$869$	−2.22471e6	−0.0999366
$870$	0	0
$871$	5.11327e6	0.228378
$872$	8.10431e6	0.360932
$873$	7.50530e7	3.33298
$874$	−514419.	−0.0227792
$875$	0	0
$876$	−2.77696e7	−1.22267
$877$	8.07539e6	0.354539	0.177270	−	0.984162i	$-0.443274\pi$
$877$	8.07539e6	0.354539	0.177270	+	0.984162i	$0.443274\pi$
$878$	−1.86315e7	−0.815665
$879$	3.57979e7	1.56273
$880$	0	0
$881$	−5.24166e6	−0.227525	−0.113763	−	0.993508i	$-0.536290\pi$
$881$	−5.24166e6	−0.227525	−0.113763	+	0.993508i	$0.536290\pi$
$882$	1.93683e7	0.838340
$883$	4.02805e7	1.73857	0.869287	−	0.494308i	$-0.164578\pi$
$883$	4.02805e7	1.73857	0.869287	+	0.494308i	$0.164578\pi$
$884$	−1.21845e7	−0.524418
$885$	0	0
$886$	−1.99880e7	−0.855431
$887$	−1.38423e7	−0.590745	−0.295373	−	0.955382i	$-0.595444\pi$
$887$	−1.38423e7	−0.590745	−0.295373	+	0.955382i	$0.595444\pi$
$888$	65045.5	0.00276812
$889$	−1.08749e7	−0.461499
$890$	0	0
$891$	−1.58250e7	−0.667807
$892$	−1.15227e6	−0.0484888
$893$	1.04643e7	0.439118
$894$	−4.14755e7	−1.73559
$895$	0	0
$896$	−1.49882e6	−0.0623706
$897$	1.00661e6	0.0417715
$898$	−663076.	−0.0274393
$899$	−2.48439e7	−1.02523
$900$	0	0
$901$	4.14672e7	1.70174
$902$	4.46357e6	0.182669
$903$	−3.01107e7	−1.22886
$904$	8.44501e6	0.343700
$905$	0	0
$906$	2.58643e7	1.04684
$907$	5.53908e6	0.223573	0.111786	−	0.993732i	$-0.464343\pi$
$907$	5.53908e6	0.223573	0.111786	+	0.993732i	$0.464343\pi$
$908$	−1.81184e7	−0.729298
$909$	−1.55591e7	−0.624562
$910$	0	0
$911$	−1.76513e7	−0.704660	−0.352330	−	0.935876i	$-0.614611\pi$
$911$	−1.76513e7	−0.704660	−0.352330	+	0.935876i	$0.614611\pi$
$912$	1.23907e7	0.493297
$913$	−7.17795e6	−0.284986
$914$	−3.22903e7	−1.27852
$915$	0	0
$916$	−550779.	−0.0216890
$917$	1.88391e7	0.739837
$918$	−6.20979e7	−2.43204
$919$	−2.80718e7	−1.09643	−0.548215	−	0.836338i	$-0.684692\pi$
$919$	−2.80718e7	−1.09643	−0.548215	+	0.836338i	$0.684692\pi$
$920$	0	0
$921$	7.71145e7	2.99562
$922$	2.35005e7	0.910436
$923$	−2.17824e7	−0.841591
$924$	5.06173e6	0.195038
$925$	0	0
$926$	−4.09156e6	−0.156806
$927$	7.34660e6	0.280794
$928$	3.01294e6	0.114847
$929$	−3.51046e6	−0.133452	−0.0667259	−	0.997771i	$-0.521255\pi$
$929$	−3.51046e6	−0.133452	−0.0667259	+	0.997771i	$0.521255\pi$
$930$	0	0
$931$	1.42904e7	0.540345
$932$	−1.10353e7	−0.416145
$933$	4.98361e7	1.87430
$934$	−2.46881e7	−0.926019
$935$	0	0
$936$	−1.70330e7	−0.635478
$937$	3.60265e7	1.34052	0.670260	−	0.742126i	$-0.266182\pi$
$937$	3.60265e7	1.34052	0.670260	+	0.742126i	$0.266182\pi$
$938$	−4.03420e6	−0.149710
$939$	−3.28780e7	−1.21686
$940$	0	0
$941$	2.51917e7	0.927435	0.463717	−	0.885983i	$-0.346515\pi$
$941$	2.51917e7	0.927435	0.463717	+	0.885983i	$0.346515\pi$
$942$	−4.48144e7	−1.64547
$943$	−700329.	−0.0256462
$944$	1.04414e7	0.381355
$945$	0	0
$946$	5.57406e6	0.202509
$947$	−2.44815e6	−0.0887082	−0.0443541	−	0.999016i	$-0.514123\pi$
$947$	−2.44815e6	−0.0887082	−0.0443541	+	0.999016i	$0.514123\pi$
$948$	−8.40761e6	−0.303845
$949$	2.81655e7	1.01520
$950$	0	0
$951$	−3.31845e7	−1.18983
$952$	9.61316e6	0.343774
$953$	4.51778e7	1.61136	0.805681	−	0.592349i	$-0.201800\pi$
$953$	4.51778e7	1.61136	0.805681	+	0.592349i	$0.201800\pi$
$954$	5.79679e7	2.06213
$955$	0	0
$956$	4.63965e6	0.164188
$957$	−1.01751e7	−0.359137
$958$	2.80098e7	0.986042
$959$	1.79822e7	0.631387
$960$	0	0
$961$	4.26658e7	1.49029
$962$	−65972.9	−0.00229841
$963$	3.68001e7	1.27874
$964$	2.25077e7	0.780078
$965$	0	0
$966$	−794181.	−0.0273827
$967$	1.09821e7	0.377676	0.188838	−	0.982008i	$-0.439528\pi$
$967$	1.09821e7	0.377676	0.188838	+	0.982008i	$0.439528\pi$
$968$	−937024.	−0.0321412
$969$	−7.94716e7	−2.71896
$970$	0	0
$971$	8.62580e6	0.293597	0.146798	−	0.989166i	$-0.453103\pi$
$971$	8.62580e6	0.293597	0.146798	+	0.989166i	$0.453103\pi$
$972$	−2.30449e7	−0.782364
$973$	−2.73861e6	−0.0927358
$974$	1.46308e6	0.0494163
$975$	0	0
$976$	−1.88643e6	−0.0633892
$977$	−1.06350e7	−0.356451	−0.178226	−	0.983990i	$-0.557036\pi$
$977$	−1.06350e7	−0.356451	−0.178226	+	0.983990i	$0.557036\pi$
$978$	9.64641e6	0.322492
$979$	−613556.	−0.0204596
$980$	0	0
$981$	−7.26632e7	−2.41069
$982$	−2.00068e7	−0.662062
$983$	−2.66620e7	−0.880053	−0.440026	−	0.897985i	$-0.645031\pi$
$983$	−2.66620e7	−0.880053	−0.440026	+	0.897985i	$0.645031\pi$
$984$	1.68687e7	0.555384
$985$	0	0
$986$	−1.93244e7	−0.633016
$987$	1.61552e7	0.527861
$988$	−1.25673e7	−0.409592
$989$	−874565.	−0.0284316
$990$	0	0
$991$	2.56353e7	0.829190	0.414595	−	0.910006i	$-0.363923\pi$
$991$	2.56353e7	0.829190	0.414595	+	0.910006i	$0.363923\pi$
$992$	−8.64628e6	−0.278965
$993$	6.62816e7	2.13314
$994$	1.71855e7	0.551692
$995$	0	0
$996$	−2.71268e7	−0.866465
$997$	588022.	0.0187351	0.00936754	−	0.999956i	$-0.497018\pi$
$997$	588022.	0.0187351	0.00936754	+	0.999956i	$0.497018\pi$
$998$	6.54069e6	0.207873
$999$	−336228.	−0.0106591

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	550.6.a.h.1.1		2
5.2	odd	4		550.6.b.j.199.2		4
5.3	odd	4		550.6.b.j.199.3		4
5.4	even	2		22.6.a.d.1.2	✓	2
15.14	odd	2		198.6.a.k.1.2		2
20.19	odd	2		176.6.a.f.1.1		2
35.34	odd	2		1078.6.a.h.1.1		2
40.19	odd	2		704.6.a.p.1.2		2
40.29	even	2		704.6.a.k.1.1		2
55.54	odd	2		242.6.a.g.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
22.6.a.d.1.2	✓	2	5.4	even	2
176.6.a.f.1.1		2	20.19	odd	2
198.6.a.k.1.2		2	15.14	odd	2
242.6.a.g.1.2		2	55.54	odd	2
550.6.a.h.1.1		2	1.1	even	1	trivial
550.6.b.j.199.2		4	5.2	odd	4
550.6.b.j.199.3		4	5.3	odd	4
704.6.a.k.1.1		2	40.29	even	2
704.6.a.p.1.2		2	40.19	odd	2
1078.6.a.h.1.1		2	35.34	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 $\F_{q}$

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	550.6.a.h.1.1

Level	$550$
Weight	$6$
Character	550.1
Self dual	yes
Analytic conductor	$88.211$
Analytic rank	$1$
Dimension	$2$
CM	no
Inner twists	$1$