LMFDB - Embedded newform 650.6.a.o.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [650,6,Mod(1,650)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(650, 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("650.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 650 = 2 \cdot 5^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	650.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:	$104.249482878$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	$\mathbb{Q}[x]/(x^{5} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 601x^{3} + 1405x^{2} + 36840x - 60300 $ x^5 - x^4 - 601x^3 + 1405x^2 + 36840*x - 60300
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2\cdot 5 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-23.8063$ of defining polynomial
Character	$\chi$	$=$	650.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-4.00000 q^{2} +25.8063 q^{3} +16.0000 q^{4} -103.225 q^{6} -115.455 q^{7} -64.0000 q^{8} +422.964 q^{9} -428.556 q^{11} +412.900 q^{12} +169.000 q^{13} +461.820 q^{14} +256.000 q^{16} +591.766 q^{17} -1691.86 q^{18} +940.874 q^{19} -2979.47 q^{21} +1714.22 q^{22} +1122.65 q^{23} -1651.60 q^{24} -676.000 q^{26} +4644.20 q^{27} -1847.28 q^{28} +4415.91 q^{29} -2732.71 q^{31} -1024.00 q^{32} -11059.4 q^{33} -2367.06 q^{34} +6767.43 q^{36} +1402.78 q^{37} -3763.50 q^{38} +4361.26 q^{39} +3862.12 q^{41} +11917.9 q^{42} -12921.3 q^{43} -6856.90 q^{44} -4490.59 q^{46} -4948.22 q^{47} +6606.41 q^{48} -3477.11 q^{49} +15271.3 q^{51} +2704.00 q^{52} +23556.6 q^{53} -18576.8 q^{54} +7389.13 q^{56} +24280.5 q^{57} -17663.7 q^{58} +17736.3 q^{59} +42609.5 q^{61} +10930.8 q^{62} -48833.4 q^{63} +4096.00 q^{64} +44237.8 q^{66} +40115.3 q^{67} +9468.25 q^{68} +28971.4 q^{69} +44171.6 q^{71} -27069.7 q^{72} +38648.9 q^{73} -5611.13 q^{74} +15054.0 q^{76} +49479.0 q^{77} -17445.0 q^{78} +60465.1 q^{79} +17069.3 q^{81} -15448.5 q^{82} -40336.7 q^{83} -47671.5 q^{84} +51685.2 q^{86} +113958. q^{87} +27427.6 q^{88} -50305.7 q^{89} -19511.9 q^{91} +17962.4 q^{92} -70521.0 q^{93} +19792.9 q^{94} -26425.6 q^{96} +96623.4 q^{97} +13908.5 q^{98} -181264. q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 20 q^{2} + 9 q^{3} + 80 q^{4} - 36 q^{6} - 42 q^{7} - 320 q^{8} + 4 q^{9} - 213 q^{11} + 144 q^{12} + 845 q^{13} + 168 q^{14} + 1280 q^{16} + 601 q^{17} - 16 q^{18} - 567 q^{19} - 3700 q^{21} + 852 q^{22}+ \cdots - 440284 q^{99}+O(q^{100}) $ 5 * q - 20 * q^2 + 9 * q^3 + 80 * q^4 - 36 * q^6 - 42 * q^7 - 320 * q^8 + 4 * q^9 - 213 * q^11 + 144 * q^12 + 845 * q^13 + 168 * q^14 + 1280 * q^16 + 601 * q^17 - 16 * q^18 - 567 * q^19 - 3700 * q^21 + 852 * q^22 + 636 * q^23 - 576 * q^24 - 3380 * q^26 + 5283 * q^27 - 672 * q^28 + 540 * q^29 - 10828 * q^31 - 5120 * q^32 - 2227 * q^33 - 2404 * q^34 + 64 * q^36 + 12708 * q^37 + 2268 * q^38 + 1521 * q^39 - 11461 * q^41 + 14800 * q^42 + 2020 * q^43 - 3408 * q^44 - 2544 * q^46 + 35062 * q^47 + 2304 * q^48 - 7285 * q^49 + 12859 * q^51 + 13520 * q^52 + 7718 * q^53 - 21132 * q^54 + 2688 * q^56 + 76689 * q^57 - 2160 * q^58 - 44686 * q^59 + 18972 * q^61 + 43312 * q^62 - 23698 * q^63 + 20480 * q^64 + 8908 * q^66 + 98667 * q^67 + 9616 * q^68 - 32718 * q^69 + 23928 * q^71 - 256 * q^72 - 17529 * q^73 - 50832 * q^74 - 9072 * q^76 + 74234 * q^77 - 6084 * q^78 - 67676 * q^79 + 26629 * q^81 + 45844 * q^82 - 13689 * q^83 - 59200 * q^84 - 8080 * q^86 + 64204 * q^87 + 13632 * q^88 - 287231 * q^89 - 7098 * q^91 + 10176 * q^92 - 38102 * q^93 - 140248 * q^94 - 9216 * q^96 + 78846 * q^97 + 29140 * q^98 - 440284 * 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
$3$	25.8063	1.65547	0.827737	−	0.561117i	$-0.189628\pi$
$3$	25.8063	1.65547	0.827737	+	0.561117i	$0.189628\pi$
$4$	16.0000	0.500000
$5$	0	0
$5$	0	0
$6$	−103.225	−1.17060
$7$	−115.455	−0.890570	−0.445285	−	0.895389i	$-0.646898\pi$
$7$	−115.455	−0.890570	−0.445285	+	0.895389i	$0.646898\pi$
$8$	−64.0000	−0.353553
$9$	422.964	1.74059
$10$	0	0
$11$	−428.556	−1.06789	−0.533944	−	0.845520i	$-0.679291\pi$
$11$	−428.556	−1.06789	−0.533944	+	0.845520i	$0.679291\pi$
$12$	412.900	0.827737
$13$	169.000	0.277350
$13$	169.000	0.277350
$14$	461.820	0.629728
$15$	0	0
$16$	256.000	0.250000
$17$	591.766	0.496624	0.248312	−	0.968680i	$-0.420124\pi$
$17$	591.766	0.496624	0.248312	+	0.968680i	$0.420124\pi$
$18$	−1691.86	−1.23079
$19$	940.874	0.597926	0.298963	−	0.954265i	$-0.403359\pi$
$19$	940.874	0.597926	0.298963	+	0.954265i	$0.403359\pi$
$20$	0	0
$21$	−2979.47	−1.47432
$22$	1714.22	0.755111
$23$	1122.65	0.442511	0.221255	−	0.975216i	$-0.428985\pi$
$23$	1122.65	0.442511	0.221255	+	0.975216i	$0.428985\pi$
$24$	−1651.60	−0.585298
$25$	0	0
$26$	−676.000	−0.196116
$27$	4644.20	1.22603
$28$	−1847.28	−0.445285
$29$	4415.91	0.975047	0.487524	−	0.873110i	$-0.337900\pi$
$29$	4415.91	0.975047	0.487524	+	0.873110i	$0.337900\pi$
$30$	0	0
$31$	−2732.71	−0.510727	−0.255363	−	0.966845i	$-0.582195\pi$
$31$	−2732.71	−0.510727	−0.255363	+	0.966845i	$0.582195\pi$
$32$	−1024.00	−0.176777
$33$	−11059.4	−1.76786
$34$	−2367.06	−0.351166
$35$	0	0
$36$	6767.43	0.870296
$37$	1402.78	0.168456	0.0842280	−	0.996447i	$-0.473158\pi$
$37$	1402.78	0.168456	0.0842280	+	0.996447i	$0.473158\pi$
$38$	−3763.50	−0.422798
$39$	4361.26	0.459146
$40$	0	0
$41$	3862.12	0.358811	0.179405	−	0.983775i	$-0.442583\pi$
$41$	3862.12	0.358811	0.179405	+	0.983775i	$0.442583\pi$
$42$	11917.9	1.04250
$43$	−12921.3	−1.06570	−0.532851	−	0.846209i	$-0.678879\pi$
$43$	−12921.3	−1.06570	−0.532851	+	0.846209i	$0.678879\pi$
$44$	−6856.90	−0.533944
$45$	0	0
$46$	−4490.59	−0.312902
$47$	−4948.22	−0.326742	−0.163371	−	0.986565i	$-0.552237\pi$
$47$	−4948.22	−0.326742	−0.163371	+	0.986565i	$0.552237\pi$
$48$	6606.41	0.413868
$49$	−3477.11	−0.206885
$50$	0	0
$51$	15271.3	0.822148
$52$	2704.00	0.138675
$53$	23556.6	1.15192	0.575962	−	0.817477i	$-0.304628\pi$
$53$	23556.6	1.15192	0.575962	+	0.817477i	$0.304628\pi$
$54$	−18576.8	−0.866936
$55$	0	0
$56$	7389.13	0.314864
$57$	24280.5	0.989851
$58$	−17663.7	−0.689462
$59$	17736.3	0.663335	0.331667	−	0.943396i	$-0.392389\pi$
$59$	17736.3	0.663335	0.331667	+	0.943396i	$0.392389\pi$
$60$	0	0
$61$	42609.5	1.46616	0.733081	−	0.680141i	$-0.238082\pi$
$61$	42609.5	1.46616	0.733081	+	0.680141i	$0.238082\pi$
$62$	10930.8	0.361138
$63$	−48833.4	−1.55012
$64$	4096.00	0.125000
$65$	0	0
$66$	44237.8	1.25007
$67$	40115.3	1.09175	0.545874	−	0.837867i	$-0.316198\pi$
$67$	40115.3	1.09175	0.545874	+	0.837867i	$0.316198\pi$
$68$	9468.25	0.248312
$69$	28971.4	0.732565
$70$	0	0
$71$	44171.6	1.03991	0.519957	−	0.854193i	$-0.325948\pi$
$71$	44171.6	1.03991	0.519957	+	0.854193i	$0.325948\pi$
$72$	−27069.7	−0.615393
$73$	38648.9	0.848848	0.424424	−	0.905464i	$-0.360477\pi$
$73$	38648.9	0.848848	0.424424	+	0.905464i	$0.360477\pi$
$74$	−5611.13	−0.119116
$75$	0	0
$76$	15054.0	0.298963
$77$	49479.0	0.951030
$78$	−17445.0	−0.324665
$79$	60465.1	1.09003	0.545013	−	0.838428i	$-0.316525\pi$
$79$	60465.1	1.09003	0.545013	+	0.838428i	$0.316525\pi$
$80$	0	0
$81$	17069.3	0.289071
$82$	−15448.5	−0.253718
$83$	−40336.7	−0.642695	−0.321347	−	0.946961i	$-0.604136\pi$
$83$	−40336.7	−0.642695	−0.321347	+	0.946961i	$0.604136\pi$
$84$	−47671.5	−0.737158
$85$	0	0
$86$	51685.2	0.753565
$87$	113958.	1.61416
$88$	27427.6	0.377556
$89$	−50305.7	−0.673197	−0.336599	−	0.941648i	$-0.609277\pi$
$89$	−50305.7	−0.673197	−0.336599	+	0.941648i	$0.609277\pi$
$90$	0	0
$91$	−19511.9	−0.247000
$92$	17962.4	0.221255
$93$	−70521.0	−0.845495
$94$	19792.9	0.231041
$95$	0	0
$96$	−26425.6	−0.292649
$97$	96623.4	1.04268	0.521342	−	0.853348i	$-0.325431\pi$
$97$	96623.4	1.04268	0.521342	+	0.853348i	$0.325431\pi$
$98$	13908.5	0.146290
$99$	−181264.	−1.85876
$100$	0	0
$101$	−99817.0	−0.973646	−0.486823	−	0.873501i	$-0.661844\pi$
$101$	−99817.0	−0.973646	−0.486823	+	0.873501i	$0.661844\pi$
$102$	−61085.1	−0.581346
$103$	187609.	1.74245	0.871224	−	0.490885i	$-0.163326\pi$
$103$	187609.	1.74245	0.871224	+	0.490885i	$0.163326\pi$
$104$	−10816.0	−0.0980581
$105$	0	0
$106$	−94226.5	−0.814533
$107$	147044.	1.24162	0.620811	−	0.783961i	$-0.286804\pi$
$107$	147044.	1.24162	0.620811	+	0.783961i	$0.286804\pi$
$108$	74307.3	0.613016
$109$	−80111.3	−0.645844	−0.322922	−	0.946426i	$-0.604665\pi$
$109$	−80111.3	−0.645844	−0.322922	+	0.946426i	$0.604665\pi$
$110$	0	0
$111$	36200.6	0.278874
$112$	−29556.5	−0.222643
$113$	32759.3	0.241345	0.120673	−	0.992692i	$-0.461495\pi$
$113$	32759.3	0.241345	0.120673	+	0.992692i	$0.461495\pi$
$114$	−97121.9	−0.699931
$115$	0	0
$116$	70654.6	0.487524
$117$	71480.9	0.482754
$118$	−70945.2	−0.469048
$119$	−68322.4	−0.442278
$120$	0	0
$121$	22609.4	0.140387
$122$	−170438.	−1.03673
$123$	99666.8	0.594002
$124$	−43723.3	−0.255363
$125$	0	0
$126$	195333.	1.09610
$127$	333208.	1.83318	0.916592	−	0.399823i	$-0.130929\pi$
$127$	333208.	1.83318	0.916592	+	0.399823i	$0.130929\pi$
$128$	−16384.0	−0.0883883
$129$	−333451.	−1.76424
$130$	0	0
$131$	175588.	0.893956	0.446978	−	0.894545i	$-0.352500\pi$
$131$	175588.	0.893956	0.446978	+	0.894545i	$0.352500\pi$
$132$	−176951.	−0.883931
$133$	−108629.	−0.532495
$134$	−160461.	−0.771983
$135$	0	0
$136$	−37873.0	−0.175583
$137$	−250969.	−1.14240	−0.571201	−	0.820810i	$-0.693522\pi$
$137$	−250969.	−1.14240	−0.571201	+	0.820810i	$0.693522\pi$
$138$	−115885.	−0.518002
$139$	−17764.1	−0.0779840	−0.0389920	−	0.999240i	$-0.512415\pi$
$139$	−17764.1	−0.0779840	−0.0389920	+	0.999240i	$0.512415\pi$
$140$	0	0
$141$	−127695.	−0.540912
$142$	−176686.	−0.735330
$143$	−72426.0	−0.296179
$144$	108279.	0.435148
$145$	0	0
$146$	−154596.	−0.600226
$147$	−89731.4	−0.342493
$148$	22444.5	0.0842280
$149$	−255058.	−0.941181	−0.470591	−	0.882352i	$-0.655959\pi$
$149$	−255058.	−0.941181	−0.470591	+	0.882352i	$0.655959\pi$
$150$	0	0
$151$	−536801.	−1.91589	−0.957947	−	0.286947i	$-0.907360\pi$
$151$	−536801.	−1.91589	−0.957947	+	0.286947i	$0.907360\pi$
$152$	−60216.0	−0.211399
$153$	250296.	0.864420
$154$	−197916.	−0.672480
$155$	0	0
$156$	69780.2	0.229573
$157$	−292686.	−0.947660	−0.473830	−	0.880616i	$-0.657129\pi$
$157$	−292686.	−0.947660	−0.473830	+	0.880616i	$0.657129\pi$
$158$	−241860.	−0.770764
$159$	607909.	1.90698
$160$	0	0
$161$	−129615.	−0.394087
$162$	−68277.4	−0.204404
$163$	468501.	1.38115	0.690575	−	0.723260i	$-0.257357\pi$
$163$	468501.	1.38115	0.690575	+	0.723260i	$0.257357\pi$
$164$	61793.8	0.179405
$165$	0	0
$166$	161347.	0.454454
$167$	36218.4	0.100494	0.0502468	−	0.998737i	$-0.483999\pi$
$167$	36218.4	0.100494	0.0502468	+	0.998737i	$0.483999\pi$
$168$	190686.	0.521249
$169$	28561.0	0.0769231
$170$	0	0
$171$	397956.	1.04075
$172$	−206741.	−0.532851
$173$	455733.	1.15770	0.578849	−	0.815435i	$-0.303502\pi$
$173$	455733.	1.15770	0.578849	+	0.815435i	$0.303502\pi$
$174$	−455833.	−1.14139
$175$	0	0
$176$	−109710.	−0.266972
$177$	457708.	1.09813
$178$	201223.	0.476022
$179$	110354.	0.257427	0.128713	−	0.991682i	$-0.458915\pi$
$179$	110354.	0.257427	0.128713	+	0.991682i	$0.458915\pi$
$180$	0	0
$181$	−828916.	−1.88068	−0.940338	−	0.340242i	$-0.889491\pi$
$181$	−828916.	−1.88068	−0.940338	+	0.340242i	$0.889491\pi$
$182$	78047.7	0.174655
$183$	1.09959e6	2.42719
$184$	−71849.4	−0.156451
$185$	0	0
$186$	282084.	0.597855
$187$	−253605.	−0.530339
$188$	−79171.6	−0.163371
$189$	−536197.	−1.09187
$190$	0	0
$191$	164897.	0.327061	0.163530	−	0.986538i	$-0.447712\pi$
$191$	164897.	0.327061	0.163530	+	0.986538i	$0.447712\pi$
$192$	105703.	0.206934
$193$	808004.	1.56142	0.780711	−	0.624892i	$-0.214857\pi$
$193$	808004.	1.56142	0.780711	+	0.624892i	$0.214857\pi$
$194$	−386494.	−0.737289
$195$	0	0
$196$	−55633.8	−0.103442
$197$	−148038.	−0.271774	−0.135887	−	0.990724i	$-0.543388\pi$
$197$	−148038.	−0.271774	−0.135887	+	0.990724i	$0.543388\pi$
$198$	725056.	1.31434
$199$	433179.	0.775415	0.387708	−	0.921782i	$-0.373267\pi$
$199$	433179.	0.775415	0.387708	+	0.921782i	$0.373267\pi$
$200$	0	0
$201$	1.03523e6	1.80736
$202$	399268.	0.688472
$203$	−509840.	−0.868348
$204$	244340.	0.411074
$205$	0	0
$206$	−750435.	−1.23210
$207$	474840.	0.770231
$208$	43264.0	0.0693375
$209$	−403218.	−0.638519
$210$	0	0
$211$	791530.	1.22394	0.611971	−	0.790880i	$-0.290377\pi$
$211$	791530.	1.22394	0.611971	+	0.790880i	$0.290377\pi$
$212$	376906.	0.575962
$213$	1.13990e6	1.72155
$214$	−588178.	−0.877959
$215$	0	0
$216$	−297229.	−0.433468
$217$	315505.	0.454838
$218$	320445.	0.456680
$219$	997384.	1.40525
$220$	0	0
$221$	100008.	0.137739
$222$	−144803.	−0.197194
$223$	151340.	0.203794	0.101897	−	0.994795i	$-0.467509\pi$
$223$	151340.	0.203794	0.101897	+	0.994795i	$0.467509\pi$
$224$	118226.	0.157432
$225$	0	0
$226$	−131037.	−0.170657
$227$	1.33510e6	1.71968	0.859840	−	0.510563i	$-0.170563\pi$
$227$	1.33510e6	1.71968	0.859840	+	0.510563i	$0.170563\pi$
$228$	388487.	0.494926
$229$	39849.9	0.0502156	0.0251078	−	0.999685i	$-0.492007\pi$
$229$	39849.9	0.0502156	0.0251078	+	0.999685i	$0.492007\pi$
$230$	0	0
$231$	1.27687e6	1.57440
$232$	−282618.	−0.344731
$233$	902942.	1.08961	0.544804	−	0.838563i	$-0.316604\pi$
$233$	902942.	1.08961	0.544804	+	0.838563i	$0.316604\pi$
$234$	−285924.	−0.341358
$235$	0	0
$236$	283781.	0.331667
$237$	1.56038e6	1.80451
$238$	273290.	0.312738
$239$	614525.	0.695896	0.347948	−	0.937514i	$-0.386879\pi$
$239$	614525.	0.695896	0.347948	+	0.937514i	$0.386879\pi$
$240$	0	0
$241$	−104935.	−0.116379	−0.0581897	−	0.998306i	$-0.518533\pi$
$241$	−104935.	−0.116379	−0.0581897	+	0.998306i	$0.518533\pi$
$242$	−90437.7	−0.0992684
$243$	−688045.	−0.747483
$244$	681752.	0.733081
$245$	0	0
$246$	−398667.	−0.420023
$247$	159008.	0.165835
$248$	174893.	0.180569
$249$	−1.04094e6	−1.06396
$250$	0	0
$251$	1.52431e6	1.52717	0.763587	−	0.645705i	$-0.223437\pi$
$251$	1.52431e6	1.52717	0.763587	+	0.645705i	$0.223437\pi$
$252$	−781334.	−0.775060
$253$	−481118.	−0.472552
$254$	−1.33283e6	−1.29626
$255$	0	0
$256$	65536.0	0.0625000
$257$	−1.26765e6	−1.19720	−0.598599	−	0.801049i	$-0.704276\pi$
$257$	−1.26765e6	−1.19720	−0.598599	+	0.801049i	$0.704276\pi$
$258$	1.33380e6	1.24751
$259$	−161959.	−0.150022
$260$	0	0
$261$	1.86777e6	1.69716
$262$	−702351.	−0.632122
$263$	−926143.	−0.825636	−0.412818	−	0.910814i	$-0.635455\pi$
$263$	−926143.	−0.825636	−0.412818	+	0.910814i	$0.635455\pi$
$264$	707804.	0.625034
$265$	0	0
$266$	434515.	0.376531
$267$	−1.29820e6	−1.11446
$268$	641844.	0.545874
$269$	−151222.	−0.127419	−0.0637097	−	0.997968i	$-0.520293\pi$
$269$	−151222.	−0.127419	−0.0637097	+	0.997968i	$0.520293\pi$
$270$	0	0
$271$	−977358.	−0.808407	−0.404204	−	0.914669i	$-0.632451\pi$
$271$	−977358.	−0.808407	−0.404204	+	0.914669i	$0.632451\pi$
$272$	151492.	0.124156
$273$	−503530.	−0.408901
$274$	1.00388e6	0.807801
$275$	0	0
$276$	463542.	0.366282
$277$	1506.04	0.00117934	0.000589668	−	1.00000i	$-0.499812\pi$
$277$	1506.04	0.00117934	0.000589668		1.00000i	$0.499812\pi$
$278$	71056.3	0.0551430
$279$	−1.15584e6	−0.888967
$280$	0	0
$281$	1.44171e6	1.08921	0.544607	−	0.838691i	$-0.316679\pi$
$281$	1.44171e6	1.08921	0.544607	+	0.838691i	$0.316679\pi$
$282$	510781.	0.382483
$283$	−2.54980e6	−1.89252	−0.946258	−	0.323413i	$-0.895170\pi$
$283$	−2.54980e6	−1.89252	−0.946258	+	0.323413i	$0.895170\pi$
$284$	706746.	0.519957
$285$	0	0
$286$	289704.	0.209430
$287$	−445901.	−0.319546
$288$	−433115.	−0.307696
$289$	−1.06967e6	−0.753365
$290$	0	0
$291$	2.49349e6	1.72614
$292$	618382.	0.424424
$293$	−1.70676e6	−1.16146	−0.580728	−	0.814098i	$-0.697232\pi$
$293$	−1.70676e6	−1.16146	−0.580728	+	0.814098i	$0.697232\pi$
$294$	358926.	0.242179
$295$	0	0
$296$	−89778.2	−0.0595582
$297$	−1.99030e6	−1.30927
$298$	1.02023e6	0.665516
$299$	189727.	0.122730
$300$	0	0
$301$	1.49183e6	0.949082
$302$	2.14721e6	1.35474
$303$	−2.57591e6	−1.61185
$304$	240864.	0.149482
$305$	0	0
$306$	−1.00118e6	−0.611237
$307$	2.78504e6	1.68649	0.843247	−	0.537526i	$-0.180641\pi$
$307$	2.78504e6	1.68649	0.843247	+	0.537526i	$0.180641\pi$
$308$	791664.	0.475515
$309$	4.84148e6	2.88458
$310$	0	0
$311$	−3.08423e6	−1.80820	−0.904098	−	0.427325i	$-0.859456\pi$
$311$	−3.08423e6	−1.80820	−0.904098	+	0.427325i	$0.859456\pi$
$312$	−279121.	−0.162333
$313$	220186.	0.127037	0.0635183	−	0.997981i	$-0.479768\pi$
$313$	220186.	0.127037	0.0635183	+	0.997981i	$0.479768\pi$
$314$	1.17074e6	0.670097
$315$	0	0
$316$	967441.	0.545013
$317$	592873.	0.331370	0.165685	−	0.986179i	$-0.447016\pi$
$317$	592873.	0.331370	0.165685	+	0.986179i	$0.447016\pi$
$318$	−2.43164e6	−1.34844
$319$	−1.89247e6	−1.04124
$320$	0	0
$321$	3.79467e6	2.05547
$322$	518462.	0.278661
$323$	556777.	0.296944
$324$	273109.	0.144535
$325$	0	0
$326$	−1.87400e6	−0.976621
$327$	−2.06737e6	−1.06918
$328$	−247175.	−0.126859
$329$	571298.	0.290986
$330$	0	0
$331$	−198774.	−0.0997218	−0.0498609	−	0.998756i	$-0.515878\pi$
$331$	−198774.	−0.0997218	−0.0498609	+	0.998756i	$0.515878\pi$
$332$	−645386.	−0.321347
$333$	593327.	0.293213
$334$	−144874.	−0.0710597
$335$	0	0
$336$	−762744.	−0.368579
$337$	2.05554e6	0.985940	0.492970	−	0.870046i	$-0.335911\pi$
$337$	2.05554e6	0.985940	0.492970	+	0.870046i	$0.335911\pi$
$338$	−114244.	−0.0543928
$339$	845397.	0.399541
$340$	0	0
$341$	1.17112e6	0.545399
$342$	−1.59182e6	−0.735919
$343$	2.34190e6	1.07482
$344$	826964.	0.376782
$345$	0	0
$346$	−1.82293e6	−0.818616
$347$	1.92285e6	0.857279	0.428639	−	0.903476i	$-0.358993\pi$
$347$	1.92285e6	0.857279	0.428639	+	0.903476i	$0.358993\pi$
$348$	1.82333e6	0.807082
$349$	−3.55687e6	−1.56317	−0.781583	−	0.623802i	$-0.785587\pi$
$349$	−3.55687e6	−1.56317	−0.781583	+	0.623802i	$0.785587\pi$
$350$	0	0
$351$	784870.	0.340040
$352$	438842.	0.188778
$353$	751255.	0.320886	0.160443	−	0.987045i	$-0.448708\pi$
$353$	751255.	0.320886	0.160443	+	0.987045i	$0.448708\pi$
$354$	−1.83083e6	−0.776497
$355$	0	0
$356$	−804891.	−0.336599
$357$	−1.76315e6	−0.732180
$358$	−441414.	−0.182028
$359$	656399.	0.268802	0.134401	−	0.990927i	$-0.457089\pi$
$359$	656399.	0.268802	0.134401	+	0.990927i	$0.457089\pi$
$360$	0	0
$361$	−1.59085e6	−0.642484
$362$	3.31566e6	1.32984
$363$	583465.	0.232406
$364$	−312191.	−0.123500
$365$	0	0
$366$	−4.39837e6	−1.71628
$367$	1.50179e6	0.582028	0.291014	−	0.956719i	$-0.406007\pi$
$367$	1.50179e6	0.582028	0.291014	+	0.956719i	$0.406007\pi$
$368$	287398.	0.110628
$369$	1.63354e6	0.624544
$370$	0	0
$371$	−2.71973e6	−1.02587
$372$	−1.12834e6	−0.422747
$373$	292527.	0.108866	0.0544332	−	0.998517i	$-0.482665\pi$
$373$	292527.	0.108866	0.0544332	+	0.998517i	$0.482665\pi$
$374$	1.01442e6	0.375006
$375$	0	0
$376$	316686.	0.115521
$377$	746289.	0.270429
$378$	2.14479e6	0.772067
$379$	−392340.	−0.140302	−0.0701511	−	0.997536i	$-0.522348\pi$
$379$	−392340.	−0.140302	−0.0701511	+	0.997536i	$0.522348\pi$
$380$	0	0
$381$	8.59886e6	3.03479
$382$	−659587.	−0.231267
$383$	−5.32936e6	−1.85643	−0.928214	−	0.372046i	$-0.878656\pi$
$383$	−5.32936e6	−1.85643	−0.928214	+	0.372046i	$0.878656\pi$
$384$	−422810.	−0.146325
$385$	0	0
$386$	−3.23202e6	−1.10409
$387$	−5.46525e6	−1.85495
$388$	1.54597e6	0.521342
$389$	−2.95835e6	−0.991231	−0.495616	−	0.868542i	$-0.665057\pi$
$389$	−2.95835e6	−0.991231	−0.495616	+	0.868542i	$0.665057\pi$
$390$	0	0
$391$	664344.	0.219761
$392$	222535.	0.0731449
$393$	4.53127e6	1.47992
$394$	592152.	0.192173
$395$	0	0
$396$	−2.90022e6	−0.929380
$397$	−845388.	−0.269203	−0.134601	−	0.990900i	$-0.542975\pi$
$397$	−845388.	−0.269203	−0.134601	+	0.990900i	$0.542975\pi$
$398$	−1.73271e6	−0.548301
$399$	−2.80330e6	−0.881532
$400$	0	0
$401$	532221.	0.165284	0.0826420	−	0.996579i	$-0.473664\pi$
$401$	532221.	0.165284	0.0826420	+	0.996579i	$0.473664\pi$
$402$	−4.14090e6	−1.27800
$403$	−461827.	−0.141650
$404$	−1.59707e6	−0.486823
$405$	0	0
$406$	2.03936e6	0.614015
$407$	−601172.	−0.179892
$408$	−977362.	−0.290673
$409$	−90104.5	−0.0266341	−0.0133171	−	0.999911i	$-0.504239\pi$
$409$	−90104.5	−0.0266341	−0.0133171	+	0.999911i	$0.504239\pi$
$410$	0	0
$411$	−6.47659e6	−1.89122
$412$	3.00174e6	0.871224
$413$	−2.04775e6	−0.590746
$414$	−1.89936e6	−0.544636
$415$	0	0
$416$	−173056.	−0.0490290
$417$	−458425.	−0.129100
$418$	1.61287e6	0.451501
$419$	4.26897e6	1.18792	0.593962	−	0.804493i	$-0.297563\pi$
$419$	4.26897e6	1.18792	0.593962	+	0.804493i	$0.297563\pi$
$420$	0	0
$421$	241157.	0.0663124	0.0331562	−	0.999450i	$-0.489444\pi$
$421$	241157.	0.0663124	0.0331562	+	0.999450i	$0.489444\pi$
$422$	−3.16612e6	−0.865458
$423$	−2.09292e6	−0.568724
$424$	−1.50762e6	−0.407266
$425$	0	0
$426$	−4.55962e6	−1.21732
$427$	−4.91949e6	−1.30572
$428$	2.35271e6	0.620811
$429$	−1.86905e6	−0.490317
$430$	0	0
$431$	−4.37702e6	−1.13497	−0.567487	−	0.823383i	$-0.692084\pi$
$431$	−4.37702e6	−1.13497	−0.567487	+	0.823383i	$0.692084\pi$
$432$	1.18892e6	0.306508
$433$	−1.11821e6	−0.286618	−0.143309	−	0.989678i	$-0.545774\pi$
$433$	−1.11821e6	−0.286618	−0.143309	+	0.989678i	$0.545774\pi$
$434$	−1.26202e6	−0.321619
$435$	0	0
$436$	−1.28178e6	−0.322922
$437$	1.05627e6	0.264589
$438$	−3.98954e6	−0.993658
$439$	3.49497e6	0.865530	0.432765	−	0.901507i	$-0.357538\pi$
$439$	3.49497e6	0.865530	0.432765	+	0.901507i	$0.357538\pi$
$440$	0	0
$441$	−1.47069e6	−0.360102
$442$	−400034.	−0.0973959
$443$	−396387.	−0.0959645	−0.0479822	−	0.998848i	$-0.515279\pi$
$443$	−396387.	−0.0959645	−0.0479822	+	0.998848i	$0.515279\pi$
$444$	579210.	0.139437
$445$	0	0
$446$	−605359.	−0.144104
$447$	−6.58210e6	−1.55810
$448$	−472904.	−0.111321
$449$	−4.43281e6	−1.03768	−0.518839	−	0.854872i	$-0.673636\pi$
$449$	−4.43281e6	−1.03768	−0.518839	+	0.854872i	$0.673636\pi$
$450$	0	0
$451$	−1.65513e6	−0.383170
$452$	524149.	0.120673
$453$	−1.38528e7	−3.17171
$454$	−5.34038e6	−1.21600
$455$	0	0
$456$	−1.55395e6	−0.349965
$457$	8.67899e6	1.94392	0.971960	−	0.235145i	$-0.0755566\pi$
$457$	8.67899e6	1.94392	0.971960	+	0.235145i	$0.0755566\pi$
$458$	−159400.	−0.0355078
$459$	2.74828e6	0.608877
$460$	0	0
$461$	−4.02089e6	−0.881191	−0.440596	−	0.897706i	$-0.645233\pi$
$461$	−4.02089e6	−0.881191	−0.440596	+	0.897706i	$0.645233\pi$
$462$	−5.10748e6	−1.11327
$463$	−4.87552e6	−1.05698	−0.528492	−	0.848938i	$-0.677243\pi$
$463$	−4.87552e6	−1.05698	−0.528492	+	0.848938i	$0.677243\pi$
$464$	1.13047e6	0.243762
$465$	0	0
$466$	−3.61177e6	−0.770469
$467$	2.69785e6	0.572434	0.286217	−	0.958165i	$-0.407602\pi$
$467$	2.69785e6	0.572434	0.286217	+	0.958165i	$0.407602\pi$
$468$	1.14369e6	0.241377
$469$	−4.63151e6	−0.972278
$470$	0	0
$471$	−7.55313e6	−1.56883
$472$	−1.13512e6	−0.234524
$473$	5.53751e6	1.13805
$474$	−6.24151e6	−1.27598
$475$	0	0
$476$	−1.09316e6	−0.221139
$477$	9.96361e6	2.00503
$478$	−2.45810e6	−0.492073
$479$	−3.68308e6	−0.733452	−0.366726	−	0.930329i	$-0.619521\pi$
$479$	−3.68308e6	−0.733452	−0.366726	+	0.930329i	$0.619521\pi$
$480$	0	0
$481$	237070.	0.0467213
$482$	419738.	0.0822926
$483$	−3.34489e6	−0.652400
$484$	361751.	0.0701934
$485$	0	0
$486$	2.75218e6	0.528550
$487$	7.41317e6	1.41639	0.708193	−	0.706019i	$-0.249511\pi$
$487$	7.41317e6	1.41639	0.708193	+	0.706019i	$0.249511\pi$
$488$	−2.72701e6	−0.518367
$489$	1.20903e7	2.28646
$490$	0	0
$491$	−590049.	−0.110455	−0.0552274	−	0.998474i	$-0.517588\pi$
$491$	−590049.	−0.110455	−0.0552274	+	0.998474i	$0.517588\pi$
$492$	1.59467e6	0.297001
$493$	2.61319e6	0.484232
$494$	−636031.	−0.117263
$495$	0	0
$496$	−699573.	−0.127682
$497$	−5.09984e6	−0.926116
$498$	4.16376e6	0.752336
$499$	6.92103e6	1.24428	0.622142	−	0.782905i	$-0.286263\pi$
$499$	6.92103e6	1.24428	0.622142	+	0.782905i	$0.286263\pi$
$500$	0	0
$501$	934663.	0.166365
$502$	−6.09723e6	−1.07987
$503$	−2.97021e6	−0.523441	−0.261721	−	0.965144i	$-0.584290\pi$
$503$	−2.97021e6	−0.523441	−0.261721	+	0.965144i	$0.584290\pi$
$504$	3.12534e6	0.548050
$505$	0	0
$506$	1.92447e6	0.334145
$507$	737053.	0.127344
$508$	5.33133e6	0.916592
$509$	−1.37177e6	−0.234687	−0.117343	−	0.993091i	$-0.537438\pi$
$509$	−1.37177e6	−0.234687	−0.117343	+	0.993091i	$0.537438\pi$
$510$	0	0
$511$	−4.46221e6	−0.755958
$512$	−262144.	−0.0441942
$513$	4.36961e6	0.733077
$514$	5.07059e6	0.846547
$515$	0	0
$516$	−5.33522e6	−0.882120
$517$	2.12059e6	0.348924
$518$	647834.	0.106081
$519$	1.17608e7	1.91654
$520$	0	0
$521$	−3.40898e6	−0.550213	−0.275106	−	0.961414i	$-0.588713\pi$
$521$	−3.40898e6	−0.550213	−0.275106	+	0.961414i	$0.588713\pi$
$522$	−7.47109e6	−1.20007
$523$	−4.90440e6	−0.784028	−0.392014	−	0.919959i	$-0.628222\pi$
$523$	−4.90440e6	−0.784028	−0.392014	+	0.919959i	$0.628222\pi$
$524$	2.80940e6	0.446978
$525$	0	0
$526$	3.70457e6	0.583813
$527$	−1.61712e6	−0.253639
$528$	−2.83122e6	−0.441965
$529$	−5.17601e6	−0.804184
$530$	0	0
$531$	7.50181e6	1.15460
$532$	−1.73806e6	−0.266248
$533$	652697.	0.0995162
$534$	5.19281e6	0.788042
$535$	0	0
$536$	−2.56738e6	−0.385991
$537$	2.84782e6	0.426163
$538$	604889.	0.0900991
$539$	1.49014e6	0.220930
$540$	0	0
$541$	−5.52866e6	−0.812132	−0.406066	−	0.913844i	$-0.633100\pi$
$541$	−5.52866e6	−0.812132	−0.406066	+	0.913844i	$0.633100\pi$
$542$	3.90943e6	0.571630
$543$	−2.13912e7	−3.11341
$544$	−605968.	−0.0877915
$545$	0	0
$546$	2.01412e6	0.289137
$547$	−9.53313e6	−1.36228	−0.681141	−	0.732152i	$-0.738516\pi$
$547$	−9.53313e6	−1.36228	−0.681141	+	0.732152i	$0.738516\pi$
$548$	−4.01551e6	−0.571201
$549$	1.80223e7	2.55199
$550$	0	0
$551$	4.15482e6	0.583006
$552$	−1.85417e6	−0.259001
$553$	−6.98100e6	−0.970744
$554$	−6024.17	−0.000833917 0
$555$	0	0
$556$	−284225.	−0.0389920
$557$	7.40068e6	1.01073	0.505363	−	0.862907i	$-0.331359\pi$
$557$	7.40068e6	1.01073	0.505363	+	0.862907i	$0.331359\pi$
$558$	4.62334e6	0.628595
$559$	−2.18370e6	−0.295572
$560$	0	0
$561$	−6.54460e6	−0.877962
$562$	−5.76686e6	−0.770191
$563$	4.46273e6	0.593375	0.296688	−	0.954975i	$-0.404118\pi$
$563$	4.46273e6	0.593375	0.296688	+	0.954975i	$0.404118\pi$
$564$	−2.04312e6	−0.270456
$565$	0	0
$566$	1.01992e7	1.33821
$567$	−1.97074e6	−0.257438
$568$	−2.82698e6	−0.367665
$569$	886521.	0.114791	0.0573956	−	0.998352i	$-0.481720\pi$
$569$	886521.	0.114791	0.0573956	+	0.998352i	$0.481720\pi$
$570$	0	0
$571$	1.14109e7	1.46464	0.732320	−	0.680961i	$-0.238438\pi$
$571$	1.14109e7	1.46464	0.732320	+	0.680961i	$0.238438\pi$
$572$	−1.15882e6	−0.148090
$573$	4.25537e6	0.541441
$574$	1.78360e6	0.225953
$575$	0	0
$576$	1.73246e6	0.217574
$577$	1.12502e7	1.40676	0.703381	−	0.710813i	$-0.251673\pi$
$577$	1.12502e7	1.40676	0.703381	+	0.710813i	$0.251673\pi$
$578$	4.27868e6	0.532709
$579$	2.08516e7	2.58489
$580$	0	0
$581$	4.65707e6	0.572365
$582$	−9.97396e6	−1.22056
$583$	−1.00953e7	−1.23013
$584$	−2.47353e6	−0.300113
$585$	0	0
$586$	6.82703e6	0.821274
$587$	−1.14799e7	−1.37513	−0.687564	−	0.726124i	$-0.741320\pi$
$587$	−1.14799e7	−1.37513	−0.687564	+	0.726124i	$0.741320\pi$
$588$	−1.43570e6	−0.171246
$589$	−2.57113e6	−0.305377
$590$	0	0
$591$	−3.82031e6	−0.449915
$592$	359113.	0.0421140
$593$	−1.53809e7	−1.79616	−0.898081	−	0.439831i	$-0.855039\pi$
$593$	−1.53809e7	−1.79616	−0.898081	+	0.439831i	$0.855039\pi$
$594$	7.96121e6	0.925791
$595$	0	0
$596$	−4.08093e6	−0.470591
$597$	1.11787e7	1.28368
$598$	−758910.	−0.0867835
$599$	−1.38707e7	−1.57954	−0.789772	−	0.613400i	$-0.789801\pi$
$599$	−1.38707e7	−1.57954	−0.789772	+	0.613400i	$0.789801\pi$
$600$	0	0
$601$	5.47553e6	0.618358	0.309179	−	0.951004i	$-0.399946\pi$
$601$	5.47553e6	0.618358	0.309179	+	0.951004i	$0.399946\pi$
$602$	−5.96733e6	−0.671102
$603$	1.69673e7	1.90029
$604$	−8.58882e6	−0.957947
$605$	0	0
$606$	1.03036e7	1.13975
$607$	9.77961e6	1.07733	0.538666	−	0.842519i	$-0.318928\pi$
$607$	9.77961e6	1.07733	0.538666	+	0.842519i	$0.318928\pi$
$608$	−963455.	−0.105699
$609$	−1.31571e7	−1.43753
$610$	0	0
$611$	−836250.	−0.0906219
$612$	4.00473e6	0.432210
$613$	−4.09960e6	−0.440647	−0.220323	−	0.975427i	$-0.570711\pi$
$613$	−4.09960e6	−0.440647	−0.220323	+	0.975427i	$0.570711\pi$
$614$	−1.11401e7	−1.19253
$615$	0	0
$616$	−3.16666e6	−0.336240
$617$	−6.05512e6	−0.640339	−0.320169	−	0.947360i	$-0.603740\pi$
$617$	−6.05512e6	−0.640339	−0.320169	+	0.947360i	$0.603740\pi$
$618$	−1.93659e7	−2.03970
$619$	1.35933e7	1.42593	0.712963	−	0.701202i	$-0.247353\pi$
$619$	1.35933e7	1.42593	0.712963	+	0.701202i	$0.247353\pi$
$620$	0	0
$621$	5.21380e6	0.542532
$622$	1.23369e7	1.27859
$623$	5.80805e6	0.599529
$624$	1.11648e6	0.114786
$625$	0	0
$626$	−880744.	−0.0898284
$627$	−1.04055e7	−1.05705
$628$	−4.68297e6	−0.473830
$629$	830119.	0.0836592
$630$	0	0
$631$	1.28129e7	1.28107	0.640536	−	0.767928i	$-0.278712\pi$
$631$	1.28129e7	1.28107	0.640536	+	0.767928i	$0.278712\pi$
$632$	−3.86976e6	−0.385382
$633$	2.04264e7	2.02620
$634$	−2.37149e6	−0.234314
$635$	0	0
$636$	9.72654e6	0.953489
$637$	−587632.	−0.0573796
$638$	7.56987e6	0.736269
$639$	1.86830e7	1.81007
$640$	0	0
$641$	−9.80825e6	−0.942858	−0.471429	−	0.881904i	$-0.656262\pi$
$641$	−9.80825e6	−0.942858	−0.471429	+	0.881904i	$0.656262\pi$
$642$	−1.51787e7	−1.45344
$643$	−3.19540e6	−0.304788	−0.152394	−	0.988320i	$-0.548698\pi$
$643$	−3.19540e6	−0.304788	−0.152394	+	0.988320i	$0.548698\pi$
$644$	−2.07385e6	−0.197043
$645$	0	0
$646$	−2.22711e6	−0.209971
$647$	−7.34835e6	−0.690127	−0.345063	−	0.938579i	$-0.612143\pi$
$647$	−7.34835e6	−0.690127	−0.345063	+	0.938579i	$0.612143\pi$
$648$	−1.09244e6	−0.102202
$649$	−7.60100e6	−0.708368
$650$	0	0
$651$	8.14201e6	0.752972
$652$	7.49601e6	0.690575
$653$	1.82689e6	0.167660	0.0838302	−	0.996480i	$-0.473285\pi$
$653$	1.82689e6	0.167660	0.0838302	+	0.996480i	$0.473285\pi$
$654$	8.26949e6	0.756022
$655$	0	0
$656$	988701.	0.0897027
$657$	1.63471e7	1.47750
$658$	−2.28519e6	−0.205758
$659$	−1.71962e7	−1.54248	−0.771240	−	0.636545i	$-0.780363\pi$
$659$	−1.71962e7	−1.54248	−0.771240	+	0.636545i	$0.780363\pi$
$660$	0	0
$661$	6.74273e6	0.600250	0.300125	−	0.953900i	$-0.402972\pi$
$661$	6.74273e6	0.600250	0.300125	+	0.953900i	$0.402972\pi$
$662$	795097.	0.0705140
$663$	2.58085e6	0.228023
$664$	2.58155e6	0.227227
$665$	0	0
$666$	−2.37331e6	−0.207333
$667$	4.95751e6	0.431469
$668$	579495.	0.0502468
$669$	3.90551e6	0.337375
$670$	0	0
$671$	−1.82606e7	−1.56570
$672$	3.05097e6	0.260625
$673$	4.64492e6	0.395313	0.197656	−	0.980271i	$-0.436667\pi$
$673$	4.64492e6	0.395313	0.197656	+	0.980271i	$0.436667\pi$
$674$	−8.22215e6	−0.697165
$675$	0	0
$676$	456976.	0.0384615
$677$	−2.10307e7	−1.76353	−0.881763	−	0.471692i	$-0.843643\pi$
$677$	−2.10307e7	−1.76353	−0.881763	+	0.471692i	$0.843643\pi$
$678$	−3.38159e6	−0.282518
$679$	−1.11557e7	−0.928584
$680$	0	0
$681$	3.44538e7	2.84689
$682$	−4.68447e6	−0.385656
$683$	−1.64815e7	−1.35190	−0.675951	−	0.736946i	$-0.736267\pi$
$683$	−1.64815e7	−1.35190	−0.675951	+	0.736946i	$0.736267\pi$
$684$	6.36730e6	0.520373
$685$	0	0
$686$	−9.36762e6	−0.760009
$687$	1.02838e6	0.0831306
$688$	−3.30786e6	−0.266425
$689$	3.98107e6	0.319486
$690$	0	0
$691$	−3.48368e6	−0.277551	−0.138776	−	0.990324i	$-0.544317\pi$
$691$	−3.48368e6	−0.277551	−0.138776	+	0.990324i	$0.544317\pi$
$692$	7.29173e6	0.578849
$693$	2.09278e7	1.65536
$694$	−7.69141e6	−0.606188
$695$	0	0
$696$	−7.29333e6	−0.570693
$697$	2.28547e6	0.178194
$698$	1.42275e7	1.10532
$699$	2.33016e7	1.80382
$700$	0	0
$701$	2.31640e7	1.78041	0.890203	−	0.455565i	$-0.150563\pi$
$701$	2.31640e7	1.78041	0.890203	+	0.455565i	$0.150563\pi$
$702$	−3.13948e6	−0.240445
$703$	1.31984e6	0.100724
$704$	−1.75537e6	−0.133486
$705$	0	0
$706$	−3.00502e6	−0.226901
$707$	1.15244e7	0.867100
$708$	7.32332e6	0.549067
$709$	−963592.	−0.0719909	−0.0359954	−	0.999352i	$-0.511460\pi$
$709$	−963592.	−0.0719909	−0.0359954	+	0.999352i	$0.511460\pi$
$710$	0	0
$711$	2.55745e7	1.89729
$712$	3.21957e6	0.238011
$713$	−3.06786e6	−0.226002
$714$	7.05259e6	0.517730
$715$	0	0
$716$	1.76566e6	0.128713
$717$	1.58586e7	1.15204
$718$	−2.62560e6	−0.190072
$719$	6.56134e6	0.473337	0.236669	−	0.971590i	$-0.423944\pi$
$719$	6.56134e6	0.473337	0.236669	+	0.971590i	$0.423944\pi$
$720$	0	0
$721$	−2.16604e7	−1.55177
$722$	6.36342e6	0.454305
$723$	−2.70797e6	−0.192663
$724$	−1.32627e7	−0.940338
$725$	0	0
$726$	−2.33386e6	−0.164336
$727$	2.35263e7	1.65089	0.825443	−	0.564485i	$-0.190925\pi$
$727$	2.35263e7	1.65089	0.825443	+	0.564485i	$0.190925\pi$
$728$	1.24876e6	0.0873276
$729$	−2.19037e7	−1.52651
$730$	0	0
$731$	−7.64639e6	−0.529253
$732$	1.75935e7	1.21360
$733$	1.73760e7	1.19451	0.597254	−	0.802052i	$-0.296258\pi$
$733$	1.73760e7	1.19451	0.597254	+	0.802052i	$0.296258\pi$
$734$	−6.00716e6	−0.411556
$735$	0	0
$736$	−1.14959e6	−0.0782256
$737$	−1.71916e7	−1.16587
$738$	−6.53414e6	−0.441619
$739$	2.38287e7	1.60505	0.802527	−	0.596616i	$-0.203488\pi$
$739$	2.38287e7	1.60505	0.802527	+	0.596616i	$0.203488\pi$
$740$	0	0
$741$	4.10340e6	0.274535
$742$	1.08789e7	0.725398
$743$	−4.87711e6	−0.324109	−0.162054	−	0.986782i	$-0.551812\pi$
$743$	−4.87711e6	−0.324109	−0.162054	+	0.986782i	$0.551812\pi$
$744$	4.51334e6	0.298927
$745$	0	0
$746$	−1.17011e6	−0.0769801
$747$	−1.70610e7	−1.11867
$748$	−4.05768e6	−0.265170
$749$	−1.69770e7	−1.10575
$750$	0	0
$751$	−9.47519e6	−0.613039	−0.306519	−	0.951864i	$-0.599164\pi$
$751$	−9.47519e6	−0.613039	−0.306519	+	0.951864i	$0.599164\pi$
$752$	−1.26675e6	−0.0816854
$753$	3.93367e7	2.52820
$754$	−2.98516e6	−0.191222
$755$	0	0
$756$	−8.57915e6	−0.545934
$757$	4.56663e6	0.289638	0.144819	−	0.989458i	$-0.453740\pi$
$757$	4.56663e6	0.289638	0.144819	+	0.989458i	$0.453740\pi$
$758$	1.56936e6	0.0992086
$759$	−1.24159e7	−0.782298
$760$	0	0
$761$	−2.29761e7	−1.43818	−0.719092	−	0.694915i	$-0.755442\pi$
$761$	−2.29761e7	−1.43818	−0.719092	+	0.694915i	$0.755442\pi$
$762$	−3.43954e7	−2.14592
$763$	9.24926e6	0.575169
$764$	2.63835e6	0.163530
$765$	0	0
$766$	2.13174e7	1.31269
$767$	2.99743e6	0.183976
$768$	1.69124e6	0.103467
$769$	2.16413e7	1.31968	0.659838	−	0.751408i	$-0.270625\pi$
$769$	2.16413e7	1.31968	0.659838	+	0.751408i	$0.270625\pi$
$770$	0	0
$771$	−3.27133e7	−1.98193
$772$	1.29281e7	0.780711
$773$	2.13234e7	1.28354	0.641768	−	0.766899i	$-0.278201\pi$
$773$	2.13234e7	1.28354	0.641768	+	0.766899i	$0.278201\pi$
$774$	2.18610e7	1.31165
$775$	0	0
$776$	−6.18390e6	−0.368645
$777$	−4.17955e6	−0.248357
$778$	1.18334e7	0.700906
$779$	3.63377e6	0.214542
$780$	0	0
$781$	−1.89300e7	−1.11051
$782$	−2.65738e6	−0.155395
$783$	2.05084e7	1.19544
$784$	−890141.	−0.0517212
$785$	0	0
$786$	−1.81251e7	−1.04646
$787$	−1.46582e7	−0.843611	−0.421806	−	0.906686i	$-0.638604\pi$
$787$	−1.46582e7	−0.843611	−0.421806	+	0.906686i	$0.638604\pi$
$788$	−2.36861e6	−0.135887
$789$	−2.39003e7	−1.36682
$790$	0	0
$791$	−3.78223e6	−0.214935
$792$	1.16009e7	0.657171
$793$	7.20101e6	0.406640
$794$	3.38155e6	0.190355
$795$	0	0
$796$	6.93086e6	0.387708
$797$	−2.77885e7	−1.54960	−0.774800	−	0.632207i	$-0.782149\pi$
$797$	−2.77885e7	−1.54960	−0.774800	+	0.632207i	$0.782149\pi$
$798$	1.12132e7	0.623337
$799$	−2.92819e6	−0.162268
$800$	0	0
$801$	−2.12775e7	−1.17176
$802$	−2.12888e6	−0.116873
$803$	−1.65632e7	−0.906475
$804$	1.65636e7	0.903680
$805$	0	0
$806$	1.84731e6	0.100162
$807$	−3.90249e6	−0.210939
$808$	6.38829e6	0.344236
$809$	−1.60118e7	−0.860140	−0.430070	−	0.902796i	$-0.641511\pi$
$809$	−1.60118e7	−0.860140	−0.430070	+	0.902796i	$0.641511\pi$
$810$	0	0
$811$	2.48699e7	1.32777	0.663883	−	0.747836i	$-0.268907\pi$
$811$	2.48699e7	1.32777	0.663883	+	0.747836i	$0.268907\pi$
$812$	−8.15744e6	−0.434174
$813$	−2.52220e7	−1.33830
$814$	2.40469e6	0.127203
$815$	0	0
$816$	3.90945e6	0.205537
$817$	−1.21573e7	−0.637211
$818$	360418.	0.0188332
$819$	−8.25284e6	−0.429926
$820$	0	0
$821$	−1.36742e7	−0.708020	−0.354010	−	0.935242i	$-0.615182\pi$
$821$	−1.36742e7	−0.708020	−0.354010	+	0.935242i	$0.615182\pi$
$822$	2.59063e7	1.33729
$823$	−1.18695e7	−0.610846	−0.305423	−	0.952217i	$-0.598798\pi$
$823$	−1.18695e7	−0.610846	−0.305423	+	0.952217i	$0.598798\pi$
$824$	−1.20070e7	−0.616049
$825$	0	0
$826$	8.19098e6	0.417721
$827$	−1.66144e7	−0.844738	−0.422369	−	0.906424i	$-0.638801\pi$
$827$	−1.66144e7	−0.844738	−0.422369	+	0.906424i	$0.638801\pi$
$828$	7.59743e6	0.385116
$829$	1.39935e7	0.707194	0.353597	−	0.935398i	$-0.384958\pi$
$829$	1.39935e7	0.707194	0.353597	+	0.935398i	$0.384958\pi$
$830$	0	0
$831$	38865.3	0.00195236
$832$	692224.	0.0346688
$833$	−2.05764e6	−0.102744
$834$	1.83370e6	0.0912878
$835$	0	0
$836$	−6.45148e6	−0.319259
$837$	−1.26912e7	−0.626167
$838$	−1.70759e7	−0.839989
$839$	1.58469e6	0.0777213	0.0388606	−	0.999245i	$-0.487627\pi$
$839$	1.58469e6	0.0777213	0.0388606	+	0.999245i	$0.487627\pi$
$840$	0	0
$841$	−1.01086e6	−0.0492833
$842$	−964629.	−0.0468900
$843$	3.72053e7	1.80317
$844$	1.26645e7	0.611971
$845$	0	0
$846$	8.37168e6	0.402149
$847$	−2.61037e6	−0.125024
$848$	6.03050e6	0.287981
$849$	−6.58008e7	−3.13301
$850$	0	0
$851$	1.57483e6	0.0745436
$852$	1.82385e7	0.860775
$853$	−4.22437e7	−1.98788	−0.993939	−	0.109933i	$-0.964936\pi$
$853$	−4.22437e7	−1.98788	−0.993939	+	0.109933i	$0.964936\pi$
$854$	1.96779e7	0.923284
$855$	0	0
$856$	−9.41084e6	−0.438979
$857$	3.17698e7	1.47762	0.738809	−	0.673915i	$-0.235388\pi$
$857$	3.17698e7	1.47762	0.738809	+	0.673915i	$0.235388\pi$
$858$	7.47618e6	0.346706
$859$	−2.92427e7	−1.35218	−0.676091	−	0.736819i	$-0.736327\pi$
$859$	−2.92427e7	−1.35218	−0.676091	+	0.736819i	$0.736327\pi$
$860$	0	0
$861$	−1.15070e7	−0.529000
$862$	1.75081e7	0.802547
$863$	−407755.	−0.0186368	−0.00931842	−	0.999957i	$-0.502966\pi$
$863$	−407755.	−0.0186368	−0.00931842	+	0.999957i	$0.502966\pi$
$864$	−4.75566e6	−0.216734
$865$	0	0
$866$	4.47284e6	0.202670
$867$	−2.76042e7	−1.24718
$868$	5.04808e6	0.227419
$869$	−2.59127e7	−1.16403
$870$	0	0
$871$	6.77948e6	0.302797
$872$	5.12712e6	0.228340
$873$	4.08682e7	1.81489
$874$	−4.22508e6	−0.187093
$875$	0	0
$876$	1.59581e7	0.702623
$877$	−1.49933e7	−0.658259	−0.329130	−	0.944285i	$-0.606755\pi$
$877$	−1.49933e7	−0.658259	−0.329130	+	0.944285i	$0.606755\pi$
$878$	−1.39799e7	−0.612022
$879$	−4.40451e7	−1.92276
$880$	0	0
$881$	−3.69735e7	−1.60491	−0.802456	−	0.596711i	$-0.796474\pi$
$881$	−3.69735e7	−1.60491	−0.802456	+	0.596711i	$0.796474\pi$
$882$	5.88278e6	0.254631
$883$	5.80647e6	0.250617	0.125308	−	0.992118i	$-0.460008\pi$
$883$	5.80647e6	0.250617	0.125308	+	0.992118i	$0.460008\pi$
$884$	1.60013e6	0.0688693
$885$	0	0
$886$	1.58555e6	0.0678571
$887$	−3.12893e7	−1.33532	−0.667662	−	0.744464i	$-0.732705\pi$
$887$	−3.12893e7	−1.33532	−0.667662	+	0.744464i	$0.732705\pi$
$888$	−2.31684e6	−0.0985970
$889$	−3.84706e7	−1.63258
$890$	0	0
$891$	−7.31517e6	−0.308695
$892$	2.42144e6	0.101897
$893$	−4.65566e6	−0.195367
$894$	2.63284e7	1.10174
$895$	0	0
$896$	1.89162e6	0.0787160
$897$	4.89616e6	0.203177
$898$	1.77312e7	0.733750
$899$	−1.20674e7	−0.497983
$900$	0	0
$901$	1.39400e7	0.572072
$902$	6.62053e6	0.270942
$903$	3.84986e7	1.57118
$904$	−2.09660e6	−0.0853285
$905$	0	0
$906$	5.54114e7	2.24274
$907$	−1.46619e7	−0.591795	−0.295897	−	0.955220i	$-0.595619\pi$
$907$	−1.46619e7	−0.591795	−0.295897	+	0.955220i	$0.595619\pi$
$908$	2.13615e7	0.859840
$909$	−4.22190e7	−1.69472
$910$	0	0
$911$	2.48498e7	0.992036	0.496018	−	0.868312i	$-0.334795\pi$
$911$	2.48498e7	0.992036	0.496018	+	0.868312i	$0.334795\pi$
$912$	6.21580e6	0.247463
$913$	1.72865e7	0.686326
$914$	−3.47159e7	−1.37456
$915$	0	0
$916$	637598.	0.0251078
$917$	−2.02725e7	−0.796130
$918$	−1.09931e7	−0.430541
$919$	4.44405e6	0.173576	0.0867881	−	0.996227i	$-0.472340\pi$
$919$	4.44405e6	0.173576	0.0867881	+	0.996227i	$0.472340\pi$
$920$	0	0
$921$	7.18714e7	2.79195
$922$	1.60836e7	0.623096
$923$	7.46500e6	0.288420
$924$	2.04299e7	0.787202
$925$	0	0
$926$	1.95021e7	0.747401
$927$	7.93518e7	3.03289
$928$	−4.52190e6	−0.172366
$929$	4.03619e6	0.153438	0.0767188	−	0.997053i	$-0.475556\pi$
$929$	4.03619e6	0.153438	0.0767188	+	0.997053i	$0.475556\pi$
$930$	0	0
$931$	−3.27153e6	−0.123702
$932$	1.44471e7	0.544804
$933$	−7.95925e7	−2.99342
$934$	−1.07914e7	−0.404772
$935$	0	0
$936$	−4.57478e6	−0.170679
$937$	2.90322e7	1.08027	0.540133	−	0.841579i	$-0.318374\pi$
$937$	2.90322e7	1.08027	0.540133	+	0.841579i	$0.318374\pi$
$938$	1.85260e7	0.687505
$939$	5.68218e6	0.210306
$940$	0	0
$941$	2.11813e7	0.779791	0.389896	−	0.920859i	$-0.372511\pi$
$941$	2.11813e7	0.779791	0.389896	+	0.920859i	$0.372511\pi$
$942$	3.02125e7	1.10933
$943$	4.33579e6	0.158778
$944$	4.54049e6	0.165834
$945$	0	0
$946$	−2.21500e7	−0.804723
$947$	2.15859e7	0.782161	0.391080	−	0.920357i	$-0.372101\pi$
$947$	2.15859e7	0.782161	0.391080	+	0.920357i	$0.372101\pi$
$948$	2.49660e7	0.902254
$949$	6.53166e6	0.235428
$950$	0	0
$951$	1.52998e7	0.548575
$952$	4.37263e6	0.156369
$953$	2.45061e7	0.874062	0.437031	−	0.899446i	$-0.356030\pi$
$953$	2.45061e7	0.874062	0.437031	+	0.899446i	$0.356030\pi$
$954$	−3.98544e7	−1.41777
$955$	0	0
$956$	9.83239e6	0.347948
$957$	−4.88375e7	−1.72375
$958$	1.47323e7	0.518629
$959$	2.89757e7	1.01739
$960$	0	0
$961$	−2.11615e7	−0.739158
$962$	−948282.	−0.0330369
$963$	6.21945e7	2.16116
$964$	−1.67895e6	−0.0581897
$965$	0	0
$966$	1.33796e7	0.461317
$967$	2.65790e7	0.914056	0.457028	−	0.889452i	$-0.348914\pi$
$967$	2.65790e7	0.914056	0.457028	+	0.889452i	$0.348914\pi$
$968$	−1.44700e6	−0.0496342
$969$	1.43684e7	0.491584
$970$	0	0
$971$	1.55283e7	0.528537	0.264268	−	0.964449i	$-0.414869\pi$
$971$	1.55283e7	0.528537	0.264268	+	0.964449i	$0.414869\pi$
$972$	−1.10087e7	−0.373742
$973$	2.05095e6	0.0694502
$974$	−2.96527e7	−1.00154
$975$	0	0
$976$	1.09080e7	0.366541
$977$	2.62698e7	0.880482	0.440241	−	0.897880i	$-0.354893\pi$
$977$	2.62698e7	0.880482	0.440241	+	0.897880i	$0.354893\pi$
$978$	−4.83610e7	−1.61677
$979$	2.15588e7	0.718900
$980$	0	0
$981$	−3.38842e7	−1.12415
$982$	2.36020e6	0.0781033
$983$	−4.94273e7	−1.63148	−0.815742	−	0.578416i	$-0.803671\pi$
$983$	−4.94273e7	−1.63148	−0.815742	+	0.578416i	$0.803671\pi$
$984$	−6.37868e6	−0.210011
$985$	0	0
$986$	−1.04527e7	−0.342403
$987$	1.47431e7	0.481720
$988$	2.54412e6	0.0829175
$989$	−1.45061e7	−0.471584
$990$	0	0
$991$	−2.40964e7	−0.779413	−0.389706	−	0.920939i	$-0.627424\pi$
$991$	−2.40964e7	−0.779413	−0.389706	+	0.920939i	$0.627424\pi$
$992$	2.79829e6	0.0902846
$993$	−5.12962e6	−0.165087
$994$	2.03994e7	0.654863
$995$	0	0
$996$	−1.66550e7	−0.531982
$997$	−2.47484e7	−0.788513	−0.394256	−	0.919000i	$-0.628998\pi$
$997$	−2.47484e7	−0.788513	−0.394256	+	0.919000i	$0.628998\pi$
$998$	−2.76841e7	−0.879841
$999$	6.51481e6	0.206532

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	650.6.a.o.1.5	✓	5
5.2	odd	4		650.6.b.m.599.1		10
5.3	odd	4		650.6.b.m.599.10		10
5.4	even	2		650.6.a.p.1.1	yes	5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
650.6.a.o.1.5	✓	5	1.1	even	1	trivial
650.6.a.p.1.1	yes	5	5.4	even	2
650.6.b.m.599.1		10	5.2	odd	4
650.6.b.m.599.10		10	5.3	odd	4

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:	\( N \)	\(=\)	\( 650 = 2 \cdot 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	650.a (trivial)

\(f(q)\)	\(=\)	\(q-4.00000 q^{2} +25.8063 q^{3} +16.0000 q^{4} -103.225 q^{6} -115.455 q^{7} -64.0000 q^{8} +422.964 q^{9} -428.556 q^{11} +412.900 q^{12} +169.000 q^{13} +461.820 q^{14} +256.000 q^{16} +591.766 q^{17} -1691.86 q^{18} +940.874 q^{19} -2979.47 q^{21} +1714.22 q^{22} +1122.65 q^{23} -1651.60 q^{24} -676.000 q^{26} +4644.20 q^{27} -1847.28 q^{28} +4415.91 q^{29} -2732.71 q^{31} -1024.00 q^{32} -11059.4 q^{33} -2367.06 q^{34} +6767.43 q^{36} +1402.78 q^{37} -3763.50 q^{38} +4361.26 q^{39} +3862.12 q^{41} +11917.9 q^{42} -12921.3 q^{43} -6856.90 q^{44} -4490.59 q^{46} -4948.22 q^{47} +6606.41 q^{48} -3477.11 q^{49} +15271.3 q^{51} +2704.00 q^{52} +23556.6 q^{53} -18576.8 q^{54} +7389.13 q^{56} +24280.5 q^{57} -17663.7 q^{58} +17736.3 q^{59} +42609.5 q^{61} +10930.8 q^{62} -48833.4 q^{63} +4096.00 q^{64} +44237.8 q^{66} +40115.3 q^{67} +9468.25 q^{68} +28971.4 q^{69} +44171.6 q^{71} -27069.7 q^{72} +38648.9 q^{73} -5611.13 q^{74} +15054.0 q^{76} +49479.0 q^{77} -17445.0 q^{78} +60465.1 q^{79} +17069.3 q^{81} -15448.5 q^{82} -40336.7 q^{83} -47671.5 q^{84} +51685.2 q^{86} +113958. q^{87} +27427.6 q^{88} -50305.7 q^{89} -19511.9 q^{91} +17962.4 q^{92} -70521.0 q^{93} +19792.9 q^{94} -26425.6 q^{96} +96623.4 q^{97} +13908.5 q^{98} -181264. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 20 q^{2} + 9 q^{3} + 80 q^{4} - 36 q^{6} - 42 q^{7} - 320 q^{8} + 4 q^{9} - 213 q^{11} + 144 q^{12} + 845 q^{13} + 168 q^{14} + 1280 q^{16} + 601 q^{17} - 16 q^{18} - 567 q^{19} - 3700 q^{21} + 852 q^{22}+ \cdots - 440284 q^{99}+O(q^{100}) \) 5 * q - 20 * q^2 + 9 * q^3 + 80 * q^4 - 36 * q^6 - 42 * q^7 - 320 * q^8 + 4 * q^9 - 213 * q^11 + 144 * q^12 + 845 * q^13 + 168 * q^14 + 1280 * q^16 + 601 * q^17 - 16 * q^18 - 567 * q^19 - 3700 * q^21 + 852 * q^22 + 636 * q^23 - 576 * q^24 - 3380 * q^26 + 5283 * q^27 - 672 * q^28 + 540 * q^29 - 10828 * q^31 - 5120 * q^32 - 2227 * q^33 - 2404 * q^34 + 64 * q^36 + 12708 * q^37 + 2268 * q^38 + 1521 * q^39 - 11461 * q^41 + 14800 * q^42 + 2020 * q^43 - 3408 * q^44 - 2544 * q^46 + 35062 * q^47 + 2304 * q^48 - 7285 * q^49 + 12859 * q^51 + 13520 * q^52 + 7718 * q^53 - 21132 * q^54 + 2688 * q^56 + 76689 * q^57 - 2160 * q^58 - 44686 * q^59 + 18972 * q^61 + 43312 * q^62 - 23698 * q^63 + 20480 * q^64 + 8908 * q^66 + 98667 * q^67 + 9616 * q^68 - 32718 * q^69 + 23928 * q^71 - 256 * q^72 - 17529 * q^73 - 50832 * q^74 - 9072 * q^76 + 74234 * q^77 - 6084 * q^78 - 67676 * q^79 + 26629 * q^81 + 45844 * q^82 - 13689 * q^83 - 59200 * q^84 - 8080 * q^86 + 64204 * q^87 + 13632 * q^88 - 287231 * q^89 - 7098 * q^91 + 10176 * q^92 - 38102 * q^93 - 140248 * q^94 - 9216 * q^96 + 78846 * q^97 + 29140 * q^98 - 440284 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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