LMFDB - Embedded newform 1028.6.a.a.1.20

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1028,6,Mod(1,1028)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1028, 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("1028.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 1028 = 2^{2} \cdot 257 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	1028.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:	$164.874566768$
Analytic rank:	$1$
Dimension:	$49$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.20
Character	$\chi$	$=$	1028.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-5.91999 q^{3} -46.7531 q^{5} +29.6970 q^{7} -207.954 q^{9} +O(q^{10})$	$q-5.91999 q^{3} -46.7531 q^{5} +29.6970 q^{7} -207.954 q^{9} -128.433 q^{11} -83.6789 q^{13} +276.778 q^{15} +308.362 q^{17} +1129.23 q^{19} -175.806 q^{21} -556.932 q^{23} -939.150 q^{25} +2669.64 q^{27} +5989.36 q^{29} -4581.52 q^{31} +760.321 q^{33} -1388.43 q^{35} +9193.37 q^{37} +495.379 q^{39} -5498.91 q^{41} +8289.54 q^{43} +9722.47 q^{45} +22257.4 q^{47} -15925.1 q^{49} -1825.50 q^{51} +9637.98 q^{53} +6004.62 q^{55} -6685.04 q^{57} +4952.35 q^{59} -6616.37 q^{61} -6175.61 q^{63} +3912.25 q^{65} +10677.7 q^{67} +3297.03 q^{69} -65397.9 q^{71} +42483.8 q^{73} +5559.76 q^{75} -3814.07 q^{77} +20472.6 q^{79} +34728.5 q^{81} +11365.4 q^{83} -14416.9 q^{85} -35457.0 q^{87} -76197.7 q^{89} -2485.01 q^{91} +27122.6 q^{93} -52795.0 q^{95} +143330. q^{97} +26708.1 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 49 q - 27 q^{3} - 89 q^{5} + 88 q^{7} + 3430 q^{9}+O(q^{10}) $ 49 * q - 27 * q^3 - 89 * q^5 + 88 * q^7 + 3430 * q^9	$ 49 q - 27 q^{3} - 89 q^{5} + 88 q^{7} + 3430 q^{9} - 484 q^{11} - 2922 q^{13} - 244 q^{15} - 3638 q^{17} - 3567 q^{19} - 3811 q^{21} + 3712 q^{23} + 13568 q^{25} - 8814 q^{27} - 5911 q^{29} - 3539 q^{31} - 17215 q^{33} - 10524 q^{35} - 31531 q^{37} - 14986 q^{39} - 35365 q^{41} - 44830 q^{43} - 34403 q^{45} + 12410 q^{47} + 68891 q^{49} - 48608 q^{51} - 79861 q^{53} - 34265 q^{55} - 109824 q^{57} - 51564 q^{59} - 66228 q^{61} + 24981 q^{63} - 31903 q^{65} - 80135 q^{67} + 50903 q^{69} + 162703 q^{71} - 58717 q^{73} + 207827 q^{75} + 89720 q^{77} + 125647 q^{79} + 420081 q^{81} - 28722 q^{83} - 125617 q^{85} + 65848 q^{87} + 93837 q^{89} - 66437 q^{91} - 278711 q^{93} - 83170 q^{95} - 347060 q^{97} + 7934 q^{99}+O(q^{100}) $ 49 * q - 27 * q^3 - 89 * q^5 + 88 * q^7 + 3430 * q^9 - 484 * q^11 - 2922 * q^13 - 244 * q^15 - 3638 * q^17 - 3567 * q^19 - 3811 * q^21 + 3712 * q^23 + 13568 * q^25 - 8814 * q^27 - 5911 * q^29 - 3539 * q^31 - 17215 * q^33 - 10524 * q^35 - 31531 * q^37 - 14986 * q^39 - 35365 * q^41 - 44830 * q^43 - 34403 * q^45 + 12410 * q^47 + 68891 * q^49 - 48608 * q^51 - 79861 * q^53 - 34265 * q^55 - 109824 * q^57 - 51564 * q^59 - 66228 * q^61 + 24981 * q^63 - 31903 * q^65 - 80135 * q^67 + 50903 * q^69 + 162703 * q^71 - 58717 * q^73 + 207827 * q^75 + 89720 * q^77 + 125647 * q^79 + 420081 * q^81 - 28722 * q^83 - 125617 * q^85 + 65848 * q^87 + 93837 * q^89 - 66437 * q^91 - 278711 * q^93 - 83170 * q^95 - 347060 * q^97 + 7934 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	−5.91999	−0.379768	−0.189884	−	0.981807i	$-0.560811\pi$
$3$	−5.91999	−0.379768	−0.189884	+	0.981807i	$0.560811\pi$
$4$	0	0
$5$	−46.7531	−0.836345	−0.418172	−	0.908368i	$-0.637329\pi$
$5$	−46.7531	−0.836345	−0.418172	+	0.908368i	$0.637329\pi$
$6$	0	0
$7$	29.6970	0.229070	0.114535	−	0.993419i	$-0.463462\pi$
$7$	29.6970	0.229070	0.114535	+	0.993419i	$0.463462\pi$
$8$	0	0
$9$	−207.954	−0.855776
$10$	0	0
$11$	−128.433	−0.320032	−0.160016	−	0.987114i	$-0.551155\pi$
$11$	−128.433	−0.320032	−0.160016	+	0.987114i	$0.551155\pi$
$12$	0	0
$13$	−83.6789	−0.137328	−0.0686638	−	0.997640i	$-0.521874\pi$
$13$	−83.6789	−0.137328	−0.0686638	+	0.997640i	$0.521874\pi$
$14$	0	0
$15$	276.778	0.317617
$16$	0	0
$17$	308.362	0.258785	0.129393	−	0.991593i	$-0.458697\pi$
$17$	308.362	0.258785	0.129393	+	0.991593i	$0.458697\pi$
$18$	0	0
$19$	1129.23	0.717627	0.358813	−	0.933409i	$-0.383181\pi$
$19$	1129.23	0.717627	0.358813	+	0.933409i	$0.383181\pi$
$20$	0	0
$21$	−175.806	−0.0869933
$22$	0	0
$23$	−556.932	−0.219524	−0.109762	−	0.993958i	$-0.535009\pi$
$23$	−556.932	−0.219524	−0.109762	+	0.993958i	$0.535009\pi$
$24$	0	0
$25$	−939.150	−0.300528
$26$	0	0
$27$	2669.64	0.704764
$28$	0	0
$29$	5989.36	1.32247	0.661235	−	0.750179i	$-0.270033\pi$
$29$	5989.36	1.32247	0.661235	+	0.750179i	$0.270033\pi$
$30$	0	0
$31$	−4581.52	−0.856259	−0.428130	−	0.903717i	$-0.640827\pi$
$31$	−4581.52	−0.856259	−0.428130	+	0.903717i	$0.640827\pi$
$32$	0	0
$33$	760.321	0.121538
$34$	0	0
$35$	−1388.43	−0.191581
$36$	0	0
$37$	9193.37	1.10400	0.552001	−	0.833843i	$-0.313864\pi$
$37$	9193.37	1.10400	0.552001	+	0.833843i	$0.313864\pi$
$38$	0	0
$39$	495.379	0.0521526
$40$	0	0
$41$	−5498.91	−0.510878	−0.255439	−	0.966825i	$-0.582220\pi$
$41$	−5498.91	−0.510878	−0.255439	+	0.966825i	$0.582220\pi$
$42$	0	0
$43$	8289.54	0.683690	0.341845	−	0.939756i	$-0.388948\pi$
$43$	8289.54	0.683690	0.341845	+	0.939756i	$0.388948\pi$
$44$	0	0
$45$	9722.47	0.715724
$46$	0	0
$47$	22257.4	1.46970	0.734851	−	0.678229i	$-0.237252\pi$
$47$	22257.4	1.46970	0.734851	+	0.678229i	$0.237252\pi$
$48$	0	0
$49$	−15925.1	−0.947527
$50$	0	0
$51$	−1825.50	−0.0982782
$52$	0	0
$53$	9637.98	0.471299	0.235649	−	0.971838i	$-0.424278\pi$
$53$	9637.98	0.471299	0.235649	+	0.971838i	$0.424278\pi$
$54$	0	0
$55$	6004.62	0.267657
$56$	0	0
$57$	−6685.04	−0.272531
$58$	0	0
$59$	4952.35	0.185217	0.0926085	−	0.995703i	$-0.470479\pi$
$59$	4952.35	0.185217	0.0926085	+	0.995703i	$0.470479\pi$
$60$	0	0
$61$	−6616.37	−0.227664	−0.113832	−	0.993500i	$-0.536313\pi$
$61$	−6616.37	−0.227664	−0.113832	+	0.993500i	$0.536313\pi$
$62$	0	0
$63$	−6175.61	−0.196033
$64$	0	0
$65$	3912.25	0.114853
$66$	0	0
$67$	10677.7	0.290597	0.145298	−	0.989388i	$-0.453586\pi$
$67$	10677.7	0.290597	0.145298	+	0.989388i	$0.453586\pi$
$68$	0	0
$69$	3297.03	0.0833682
$70$	0	0
$71$	−65397.9	−1.53964	−0.769818	−	0.638263i	$-0.779653\pi$
$71$	−65397.9	−1.53964	−0.769818	+	0.638263i	$0.779653\pi$
$72$	0	0
$73$	42483.8	0.933075	0.466538	−	0.884501i	$-0.345501\pi$
$73$	42483.8	0.933075	0.466538	+	0.884501i	$0.345501\pi$
$74$	0	0
$75$	5559.76	0.114131
$76$	0	0
$77$	−3814.07	−0.0733097
$78$	0	0
$79$	20472.6	0.369068	0.184534	−	0.982826i	$-0.440922\pi$
$79$	20472.6	0.369068	0.184534	+	0.982826i	$0.440922\pi$
$80$	0	0
$81$	34728.5	0.588130
$82$	0	0
$83$	11365.4	0.181088	0.0905438	−	0.995892i	$-0.471139\pi$
$83$	11365.4	0.181088	0.0905438	+	0.995892i	$0.471139\pi$
$84$	0	0
$85$	−14416.9	−0.216433
$86$	0	0
$87$	−35457.0	−0.502231
$88$	0	0
$89$	−76197.7	−1.01969	−0.509844	−	0.860267i	$-0.670297\pi$
$89$	−76197.7	−1.01969	−0.509844	+	0.860267i	$0.670297\pi$
$90$	0	0
$91$	−2485.01	−0.0314576
$92$	0	0
$93$	27122.6	0.325180
$94$	0	0
$95$	−52795.0	−0.600183
$96$	0	0
$97$	143330.	1.54671	0.773355	−	0.633973i	$-0.218577\pi$
$97$	143330.	1.54671	0.773355	+	0.633973i	$0.218577\pi$
$98$	0	0
$99$	26708.1	0.273876
$100$	0	0
$101$	−3314.70	−0.0323326	−0.0161663	−	0.999869i	$-0.505146\pi$
$101$	−3314.70	−0.0323326	−0.0161663	+	0.999869i	$0.505146\pi$
$102$	0	0
$103$	−96443.2	−0.895733	−0.447866	−	0.894101i	$-0.647816\pi$
$103$	−96443.2	−0.895733	−0.447866	+	0.894101i	$0.647816\pi$
$104$	0	0
$105$	8219.48	0.0727564
$106$	0	0
$107$	−109250.	−0.922491	−0.461245	−	0.887273i	$-0.652597\pi$
$107$	−109250.	−0.922491	−0.461245	+	0.887273i	$0.652597\pi$
$108$	0	0
$109$	47374.4	0.381924	0.190962	−	0.981597i	$-0.438839\pi$
$109$	47374.4	0.381924	0.190962	+	0.981597i	$0.438839\pi$
$110$	0	0
$111$	−54424.7	−0.419265
$112$	0	0
$113$	121442.	0.894688	0.447344	−	0.894362i	$-0.352370\pi$
$113$	121442.	0.894688	0.447344	+	0.894362i	$0.352370\pi$
$114$	0	0
$115$	26038.3	0.183598
$116$	0	0
$117$	17401.3	0.117522
$118$	0	0
$119$	9157.45	0.0592798
$120$	0	0
$121$	−144556.	−0.897579
$122$	0	0
$123$	32553.5	0.194015
$124$	0	0
$125$	190012.	1.08769
$126$	0	0
$127$	17402.6	0.0957426	0.0478713	−	0.998854i	$-0.484756\pi$
$127$	17402.6	0.0957426	0.0478713	+	0.998854i	$0.484756\pi$
$128$	0	0
$129$	−49074.0	−0.259643
$130$	0	0
$131$	196456.	1.00020	0.500100	−	0.865968i	$-0.333297\pi$
$131$	196456.	1.00020	0.500100	+	0.865968i	$0.333297\pi$
$132$	0	0
$133$	33534.8	0.164387
$134$	0	0
$135$	−124814.	−0.589426
$136$	0	0
$137$	95448.9	0.434480	0.217240	−	0.976118i	$-0.430295\pi$
$137$	95448.9	0.434480	0.217240	+	0.976118i	$0.430295\pi$
$138$	0	0
$139$	−7215.29	−0.0316750	−0.0158375	−	0.999875i	$-0.505041\pi$
$139$	−7215.29	−0.0316750	−0.0158375	+	0.999875i	$0.505041\pi$
$140$	0	0
$141$	−131763.	−0.558145
$142$	0	0
$143$	10747.1	0.0439492
$144$	0	0
$145$	−280021.	−1.10604
$146$	0	0
$147$	94276.4	0.359840
$148$	0	0
$149$	42153.7	0.155550	0.0777749	−	0.996971i	$-0.475218\pi$
$149$	42153.7	0.155550	0.0777749	+	0.996971i	$0.475218\pi$
$150$	0	0
$151$	−532414.	−1.90023	−0.950116	−	0.311896i	$-0.899036\pi$
$151$	−532414.	−1.90023	−0.950116	+	0.311896i	$0.899036\pi$
$152$	0	0
$153$	−64125.1	−0.221462
$154$	0	0
$155$	214200.	0.716128
$156$	0	0
$157$	406279.	1.31545	0.657726	−	0.753257i	$-0.271518\pi$
$157$	406279.	1.31545	0.657726	+	0.753257i	$0.271518\pi$
$158$	0	0
$159$	−57056.8	−0.178984
$160$	0	0
$161$	−16539.2	−0.0502864
$162$	0	0
$163$	165679.	0.488427	0.244213	−	0.969722i	$-0.421470\pi$
$163$	165679.	0.488427	0.244213	+	0.969722i	$0.421470\pi$
$164$	0	0
$165$	−35547.3	−0.101648
$166$	0	0
$167$	−377604.	−1.04772	−0.523860	−	0.851805i	$-0.675508\pi$
$167$	−377604.	−1.04772	−0.523860	+	0.851805i	$0.675508\pi$
$168$	0	0
$169$	−364291.	−0.981141
$170$	0	0
$171$	−234828.	−0.614128
$172$	0	0
$173$	−758827.	−1.92765	−0.963824	−	0.266540i	$-0.914120\pi$
$173$	−758827.	−1.92765	−0.963824	+	0.266540i	$0.914120\pi$
$174$	0	0
$175$	−27889.9	−0.0688419
$176$	0	0
$177$	−29317.9	−0.0703395
$178$	0	0
$179$	266642.	0.622008	0.311004	−	0.950409i	$-0.399335\pi$
$179$	266642.	0.622008	0.311004	+	0.950409i	$0.399335\pi$
$180$	0	0
$181$	−607851.	−1.37912	−0.689558	−	0.724230i	$-0.742195\pi$
$181$	−607851.	−1.37912	−0.689558	+	0.724230i	$0.742195\pi$
$182$	0	0
$183$	39168.9	0.0864596
$184$	0	0
$185$	−429818.	−0.923327
$186$	0	0
$187$	−39603.8	−0.0828196
$188$	0	0
$189$	79280.5	0.161440
$190$	0	0
$191$	−5512.87	−0.0109344	−0.00546719	−	0.999985i	$-0.501740\pi$
$191$	−5512.87	−0.0109344	−0.00546719	+	0.999985i	$0.501740\pi$
$192$	0	0
$193$	510117.	0.985772	0.492886	−	0.870094i	$-0.335942\pi$
$193$	510117.	0.985772	0.492886	+	0.870094i	$0.335942\pi$
$194$	0	0
$195$	−23160.5	−0.0436175
$196$	0	0
$197$	−186236.	−0.341900	−0.170950	−	0.985280i	$-0.554684\pi$
$197$	−186236.	−0.341900	−0.170950	+	0.985280i	$0.554684\pi$
$198$	0	0
$199$	−550421.	−0.985285	−0.492643	−	0.870232i	$-0.663969\pi$
$199$	−550421.	−0.985285	−0.492643	+	0.870232i	$0.663969\pi$
$200$	0	0
$201$	−63211.9	−0.110359
$202$	0	0
$203$	177866.	0.302938
$204$	0	0
$205$	257091.	0.427270
$206$	0	0
$207$	115816.	0.187864
$208$	0	0
$209$	−145030.	−0.229664
$210$	0	0
$211$	−466397.	−0.721191	−0.360595	−	0.932722i	$-0.617426\pi$
$211$	−466397.	−0.721191	−0.360595	+	0.932722i	$0.617426\pi$
$212$	0	0
$213$	387155.	0.584704
$214$	0	0
$215$	−387561.	−0.571800
$216$	0	0
$217$	−136057.	−0.196143
$218$	0	0
$219$	−251504.	−0.354352
$220$	0	0
$221$	−25803.4	−0.0355383
$222$	0	0
$223$	252894.	0.340547	0.170273	−	0.985397i	$-0.445535\pi$
$223$	252894.	0.340547	0.170273	+	0.985397i	$0.445535\pi$
$224$	0	0
$225$	195300.	0.257185
$226$	0	0
$227$	−1.27456e6	−1.64170	−0.820851	−	0.571143i	$-0.806500\pi$
$227$	−1.27456e6	−1.64170	−0.820851	+	0.571143i	$0.806500\pi$
$228$	0	0
$229$	622959.	0.785002	0.392501	−	0.919752i	$-0.371610\pi$
$229$	622959.	0.785002	0.392501	+	0.919752i	$0.371610\pi$
$230$	0	0
$231$	22579.3	0.0278407
$232$	0	0
$233$	695145.	0.838852	0.419426	−	0.907789i	$-0.362231\pi$
$233$	695145.	0.838852	0.419426	+	0.907789i	$0.362231\pi$
$234$	0	0
$235$	−1.04060e6	−1.22918
$236$	0	0
$237$	−121198.	−0.140160
$238$	0	0
$239$	−108787.	−0.123191	−0.0615957	−	0.998101i	$-0.519619\pi$
$239$	−108787.	−0.123191	−0.0615957	+	0.998101i	$0.519619\pi$
$240$	0	0
$241$	−1.01687e6	−1.12778	−0.563888	−	0.825851i	$-0.690695\pi$
$241$	−1.01687e6	−1.12778	−0.563888	+	0.825851i	$0.690695\pi$
$242$	0	0
$243$	−854316.	−0.928117
$244$	0	0
$245$	744547.	0.792459
$246$	0	0
$247$	−94492.7	−0.0985499
$248$	0	0
$249$	−67283.0	−0.0687712
$250$	0	0
$251$	−538853.	−0.539866	−0.269933	−	0.962879i	$-0.587001\pi$
$251$	−538853.	−0.539866	−0.269933	+	0.962879i	$0.587001\pi$
$252$	0	0
$253$	71528.2	0.0702548
$254$	0	0
$255$	85347.9	0.0821945
$256$	0	0
$257$	66049.0	0.0623783
$257$	66049.0	0.0623783
$258$	0	0
$259$	273016.	0.252894
$260$	0	0
$261$	−1.24551e6	−1.13174
$262$	0	0
$263$	−130941.	−0.116731	−0.0583656	−	0.998295i	$-0.518589\pi$
$263$	−130941.	−0.116731	−0.0583656	+	0.998295i	$0.518589\pi$
$264$	0	0
$265$	−450605.	−0.394168
$266$	0	0
$267$	451090.	0.387244
$268$	0	0
$269$	1.90173e6	1.60239	0.801196	−	0.598402i	$-0.204197\pi$
$269$	1.90173e6	1.60239	0.801196	+	0.598402i	$0.204197\pi$
$270$	0	0
$271$	−345305.	−0.285614	−0.142807	−	0.989751i	$-0.545613\pi$
$271$	−345305.	−0.285614	−0.142807	+	0.989751i	$0.545613\pi$
$272$	0	0
$273$	14711.3	0.0119466
$274$	0	0
$275$	120618.	0.0961786
$276$	0	0
$277$	−929549.	−0.727902	−0.363951	−	0.931418i	$-0.618572\pi$
$277$	−929549.	−0.727902	−0.363951	+	0.931418i	$0.618572\pi$
$278$	0	0
$279$	952744.	0.732767
$280$	0	0
$281$	−2.15456e6	−1.62777	−0.813883	−	0.581029i	$-0.802650\pi$
$281$	−2.15456e6	−1.62777	−0.813883	+	0.581029i	$0.802650\pi$
$282$	0	0
$283$	−1.01648e6	−0.754455	−0.377227	−	0.926121i	$-0.623123\pi$
$283$	−1.01648e6	−0.754455	−0.377227	+	0.926121i	$0.623123\pi$
$284$	0	0
$285$	312546.	0.227930
$286$	0	0
$287$	−163301.	−0.117027
$288$	0	0
$289$	−1.32477e6	−0.933030
$290$	0	0
$291$	−848515.	−0.587391
$292$	0	0
$293$	−656051.	−0.446446	−0.223223	−	0.974767i	$-0.571658\pi$
$293$	−656051.	−0.446446	−0.223223	+	0.974767i	$0.571658\pi$
$294$	0	0
$295$	−231537.	−0.154905
$296$	0	0
$297$	−342869.	−0.225547
$298$	0	0
$299$	46603.4	0.0301467
$300$	0	0
$301$	246175.	0.156613
$302$	0	0
$303$	19623.0	0.0122789
$304$	0	0
$305$	309336.	0.190406
$306$	0	0
$307$	331884.	0.200974	0.100487	−	0.994938i	$-0.467960\pi$
$307$	331884.	0.200974	0.100487	+	0.994938i	$0.467960\pi$
$308$	0	0
$309$	570943.	0.340170
$310$	0	0
$311$	2.04825e6	1.20083	0.600414	−	0.799689i	$-0.295002\pi$
$311$	2.04825e6	1.20083	0.600414	+	0.799689i	$0.295002\pi$
$312$	0	0
$313$	22450.8	0.0129530	0.00647652	−	0.999979i	$-0.497938\pi$
$313$	22450.8	0.0129530	0.00647652	+	0.999979i	$0.497938\pi$
$314$	0	0
$315$	288729.	0.163951
$316$	0	0
$317$	466922.	0.260974	0.130487	−	0.991450i	$-0.458346\pi$
$317$	466922.	0.260974	0.130487	+	0.991450i	$0.458346\pi$
$318$	0	0
$319$	−769230.	−0.423233
$320$	0	0
$321$	646759.	0.350332
$322$	0	0
$323$	348212.	0.185711
$324$	0	0
$325$	78587.0	0.0412707
$326$	0	0
$327$	−280456.	−0.145043
$328$	0	0
$329$	660978.	0.336664
$330$	0	0
$331$	225867.	0.113314	0.0566569	−	0.998394i	$-0.481956\pi$
$331$	225867.	0.113314	0.0566569	+	0.998394i	$0.481956\pi$
$332$	0	0
$333$	−1.91179e6	−0.944780
$334$	0	0
$335$	−499215.	−0.243039
$336$	0	0
$337$	1.52608e6	0.731985	0.365993	−	0.930618i	$-0.380730\pi$
$337$	1.52608e6	0.731985	0.365993	+	0.930618i	$0.380730\pi$
$338$	0	0
$339$	−718934.	−0.339774
$340$	0	0
$341$	588417.	0.274031
$342$	0	0
$343$	−972046.	−0.446120
$344$	0	0
$345$	−154146.	−0.0697245
$346$	0	0
$347$	−2.53109e6	−1.12845	−0.564226	−	0.825620i	$-0.690825\pi$
$347$	−2.53109e6	−1.12845	−0.564226	+	0.825620i	$0.690825\pi$
$348$	0	0
$349$	−4.31534e6	−1.89649	−0.948247	−	0.317532i	$-0.897146\pi$
$349$	−4.31534e6	−1.89649	−0.948247	+	0.317532i	$0.897146\pi$
$350$	0	0
$351$	−223393.	−0.0967835
$352$	0	0
$353$	−2.76456e6	−1.18083	−0.590417	−	0.807098i	$-0.701037\pi$
$353$	−2.76456e6	−1.18083	−0.590417	+	0.807098i	$0.701037\pi$
$354$	0	0
$355$	3.05755e6	1.28767
$356$	0	0
$357$	−54212.0	−0.0225126
$358$	0	0
$359$	3.49144e6	1.42978	0.714889	−	0.699238i	$-0.246477\pi$
$359$	3.49144e6	1.42978	0.714889	+	0.699238i	$0.246477\pi$
$360$	0	0
$361$	−1.20094e6	−0.485012
$362$	0	0
$363$	855771.	0.340872
$364$	0	0
$365$	−1.98625e6	−0.780372
$366$	0	0
$367$	−2.91947e6	−1.13146	−0.565730	−	0.824591i	$-0.691405\pi$
$367$	−2.91947e6	−1.13146	−0.565730	+	0.824591i	$0.691405\pi$
$368$	0	0
$369$	1.14352e6	0.437197
$370$	0	0
$371$	286219.	0.107960
$372$	0	0
$373$	−406331.	−0.151219	−0.0756097	−	0.997137i	$-0.524090\pi$
$373$	−406331.	−0.151219	−0.0756097	+	0.997137i	$0.524090\pi$
$374$	0	0
$375$	−1.12487e6	−0.413069
$376$	0	0
$377$	−501183.	−0.181611
$378$	0	0
$379$	−3.77232e6	−1.34900	−0.674498	−	0.738277i	$-0.735640\pi$
$379$	−3.77232e6	−1.34900	−0.674498	+	0.738277i	$0.735640\pi$
$380$	0	0
$381$	−103023.	−0.0363599
$382$	0	0
$383$	2.31181e6	0.805294	0.402647	−	0.915355i	$-0.368090\pi$
$383$	2.31181e6	0.805294	0.402647	+	0.915355i	$0.368090\pi$
$384$	0	0
$385$	178319.	0.0613122
$386$	0	0
$387$	−1.72384e6	−0.585086
$388$	0	0
$389$	−2.98398e6	−0.999819	−0.499909	−	0.866078i	$-0.666633\pi$
$389$	−2.98398e6	−0.999819	−0.499909	+	0.866078i	$0.666633\pi$
$390$	0	0
$391$	−171737.	−0.0568096
$392$	0	0
$393$	−1.16302e6	−0.379844
$394$	0	0
$395$	−957159.	−0.308668
$396$	0	0
$397$	−3.11326e6	−0.991379	−0.495689	−	0.868500i	$-0.665085\pi$
$397$	−3.11326e6	−0.991379	−0.495689	+	0.868500i	$0.665085\pi$
$398$	0	0
$399$	−198526.	−0.0624287
$400$	0	0
$401$	−4.48242e6	−1.39204	−0.696019	−	0.718023i	$-0.745047\pi$
$401$	−4.48242e6	−1.39204	−0.696019	+	0.718023i	$0.745047\pi$
$402$	0	0
$403$	383376.	0.117588
$404$	0	0
$405$	−1.62366e6	−0.491879
$406$	0	0
$407$	−1.18073e6	−0.353317
$408$	0	0
$409$	3.37399e6	0.997323	0.498662	−	0.866797i	$-0.333825\pi$
$409$	3.37399e6	0.997323	0.498662	+	0.866797i	$0.333825\pi$
$410$	0	0
$411$	−565057.	−0.165001
$412$	0	0
$413$	147070.	0.0424276
$414$	0	0
$415$	−531366.	−0.151452
$416$	0	0
$417$	42714.4	0.0120291
$418$	0	0
$419$	2.43068e6	0.676382	0.338191	−	0.941077i	$-0.390185\pi$
$419$	2.43068e6	0.676382	0.338191	+	0.941077i	$0.390185\pi$
$420$	0	0
$421$	−5.24622e6	−1.44258	−0.721292	−	0.692631i	$-0.756452\pi$
$421$	−5.24622e6	−1.44258	−0.721292	+	0.692631i	$0.756452\pi$
$422$	0	0
$423$	−4.62850e6	−1.25774
$424$	0	0
$425$	−289598.	−0.0777721
$426$	0	0
$427$	−196487.	−0.0521511
$428$	0	0
$429$	−63622.8	−0.0166905
$430$	0	0
$431$	3.82480e6	0.991780	0.495890	−	0.868385i	$-0.334842\pi$
$431$	3.82480e6	0.991780	0.495890	+	0.868385i	$0.334842\pi$
$432$	0	0
$433$	−2.01731e6	−0.517074	−0.258537	−	0.966001i	$-0.583240\pi$
$433$	−2.01731e6	−0.517074	−0.258537	+	0.966001i	$0.583240\pi$
$434$	0	0
$435$	1.65772e6	0.420038
$436$	0	0
$437$	−628904.	−0.157536
$438$	0	0
$439$	2.29057e6	0.567259	0.283630	−	0.958934i	$-0.408461\pi$
$439$	2.29057e6	0.567259	0.283630	+	0.958934i	$0.408461\pi$
$440$	0	0
$441$	3.31168e6	0.810871
$442$	0	0
$443$	−2.11081e6	−0.511023	−0.255511	−	0.966806i	$-0.582244\pi$
$443$	−2.11081e6	−0.511023	−0.255511	+	0.966806i	$0.582244\pi$
$444$	0	0
$445$	3.56248e6	0.852810
$446$	0	0
$447$	−249549.	−0.0590728
$448$	0	0
$449$	7.18943e6	1.68298	0.841490	−	0.540273i	$-0.181679\pi$
$449$	7.18943e6	1.68298	0.841490	+	0.540273i	$0.181679\pi$
$450$	0	0
$451$	706240.	0.163497
$452$	0	0
$453$	3.15189e6	0.721647
$454$	0	0
$455$	116182.	0.0263094
$456$	0	0
$457$	1.60271e6	0.358975	0.179488	−	0.983760i	$-0.442556\pi$
$457$	1.60271e6	0.358975	0.179488	+	0.983760i	$0.442556\pi$
$458$	0	0
$459$	823218.	0.182382
$460$	0	0
$461$	−8.60748e6	−1.88636	−0.943178	−	0.332287i	$-0.892180\pi$
$461$	−8.60748e6	−1.88636	−0.943178	+	0.332287i	$0.892180\pi$
$462$	0	0
$463$	2.40998e6	0.522469	0.261234	−	0.965275i	$-0.415870\pi$
$463$	2.40998e6	0.522469	0.261234	+	0.965275i	$0.415870\pi$
$464$	0	0
$465$	−1.26806e6	−0.271962
$466$	0	0
$467$	4.03780e6	0.856747	0.428373	−	0.903602i	$-0.359087\pi$
$467$	4.03780e6	0.856747	0.428373	+	0.903602i	$0.359087\pi$
$468$	0	0
$469$	317096.	0.0665669
$470$	0	0
$471$	−2.40517e6	−0.499566
$472$	0	0
$473$	−1.06465e6	−0.218803
$474$	0	0
$475$	−1.06052e6	−0.215667
$476$	0	0
$477$	−2.00425e6	−0.403326
$478$	0	0
$479$	−4.53702e6	−0.903509	−0.451754	−	0.892142i	$-0.649202\pi$
$479$	−4.53702e6	−0.903509	−0.451754	+	0.892142i	$0.649202\pi$
$480$	0	0
$481$	−769291.	−0.151610
$482$	0	0
$483$	97912.1	0.0190971
$484$	0	0
$485$	−6.70114e6	−1.29358
$486$	0	0
$487$	6.45929e6	1.23413	0.617067	−	0.786911i	$-0.288321\pi$
$487$	6.45929e6	1.23413	0.617067	+	0.786911i	$0.288321\pi$
$488$	0	0
$489$	−980821.	−0.185489
$490$	0	0
$491$	540507.	0.101181	0.0505903	−	0.998719i	$-0.483890\pi$
$491$	540507.	0.101181	0.0505903	+	0.998719i	$0.483890\pi$
$492$	0	0
$493$	1.84689e6	0.342235
$494$	0	0
$495$	−1.24868e6	−0.229055
$496$	0	0
$497$	−1.94212e6	−0.352684
$498$	0	0
$499$	−7.55560e6	−1.35837	−0.679185	−	0.733967i	$-0.737666\pi$
$499$	−7.55560e6	−1.35837	−0.679185	+	0.733967i	$0.737666\pi$
$500$	0	0
$501$	2.23541e6	0.397890
$502$	0	0
$503$	−1.01259e7	−1.78450	−0.892248	−	0.451546i	$-0.850873\pi$
$503$	−1.01259e7	−1.78450	−0.892248	+	0.451546i	$0.850873\pi$
$504$	0	0
$505$	154973.	0.0270412
$506$	0	0
$507$	2.15660e6	0.372606
$508$	0	0
$509$	3.17509e6	0.543203	0.271601	−	0.962410i	$-0.412447\pi$
$509$	3.17509e6	0.543203	0.271601	+	0.962410i	$0.412447\pi$
$510$	0	0
$511$	1.26164e6	0.213739
$512$	0	0
$513$	3.01464e6	0.505757
$514$	0	0
$515$	4.50901e6	0.749141
$516$	0	0
$517$	−2.85857e6	−0.470352
$518$	0	0
$519$	4.49225e6	0.732059
$520$	0	0
$521$	8.71113e6	1.40598	0.702991	−	0.711198i	$-0.251847\pi$
$521$	8.71113e6	1.40598	0.702991	+	0.711198i	$0.251847\pi$
$522$	0	0
$523$	−1.38758e6	−0.221822	−0.110911	−	0.993830i	$-0.535377\pi$
$523$	−1.38758e6	−0.221822	−0.110911	+	0.993830i	$0.535377\pi$
$524$	0	0
$525$	165108.	0.0261439
$526$	0	0
$527$	−1.41277e6	−0.221587
$528$	0	0
$529$	−6.12617e6	−0.951809
$530$	0	0
$531$	−1.02986e6	−0.158504
$532$	0	0
$533$	460143.	0.0701575
$534$	0	0
$535$	5.10777e6	0.771520
$536$	0	0
$537$	−1.57852e6	−0.236218
$538$	0	0
$539$	2.04530e6	0.303239
$540$	0	0
$541$	3.33182e6	0.489428	0.244714	−	0.969595i	$-0.421306\pi$
$541$	3.33182e6	0.489428	0.244714	+	0.969595i	$0.421306\pi$
$542$	0	0
$543$	3.59848e6	0.523744
$544$	0	0
$545$	−2.21490e6	−0.319420
$546$	0	0
$547$	5.55256e6	0.793460	0.396730	−	0.917935i	$-0.370145\pi$
$547$	5.55256e6	0.793460	0.396730	+	0.917935i	$0.370145\pi$
$548$	0	0
$549$	1.37590e6	0.194830
$550$	0	0
$551$	6.76337e6	0.949039
$552$	0	0
$553$	607977.	0.0845423
$554$	0	0
$555$	2.54452e6	0.350650
$556$	0	0
$557$	−7.56664e6	−1.03339	−0.516696	−	0.856169i	$-0.672838\pi$
$557$	−7.56664e6	−1.03339	−0.516696	+	0.856169i	$0.672838\pi$
$558$	0	0
$559$	−693659.	−0.0938894
$560$	0	0
$561$	234454.	0.0314522
$562$	0	0
$563$	1.37628e7	1.82994	0.914968	−	0.403527i	$-0.132216\pi$
$563$	1.37628e7	1.82994	0.914968	+	0.403527i	$0.132216\pi$
$564$	0	0
$565$	−5.67777e6	−0.748267
$566$	0	0
$567$	1.03133e6	0.134723
$568$	0	0
$569$	6.91370e6	0.895220	0.447610	−	0.894229i	$-0.352275\pi$
$569$	6.91370e6	0.895220	0.447610	+	0.894229i	$0.352275\pi$
$570$	0	0
$571$	1.27737e7	1.63956	0.819778	−	0.572681i	$-0.194097\pi$
$571$	1.27737e7	1.63956	0.819778	+	0.572681i	$0.194097\pi$
$572$	0	0
$573$	32636.2	0.00415253
$574$	0	0
$575$	523042.	0.0659731
$576$	0	0
$577$	7.43850e6	0.930134	0.465067	−	0.885275i	$-0.346030\pi$
$577$	7.43850e6	0.930134	0.465067	+	0.885275i	$0.346030\pi$
$578$	0	0
$579$	−3.01989e6	−0.374364
$580$	0	0
$581$	337518.	0.0414817
$582$	0	0
$583$	−1.23783e6	−0.150831
$584$	0	0
$585$	−813566.	−0.0982886
$586$	0	0
$587$	1.45458e7	1.74238	0.871190	−	0.490946i	$-0.163349\pi$
$587$	1.45458e7	1.74238	0.871190	+	0.490946i	$0.163349\pi$
$588$	0	0
$589$	−5.17359e6	−0.614474
$590$	0	0
$591$	1.10252e6	0.129843
$592$	0	0
$593$	7.86180e6	0.918090	0.459045	−	0.888413i	$-0.348192\pi$
$593$	7.86180e6	0.918090	0.459045	+	0.888413i	$0.348192\pi$
$594$	0	0
$595$	−428139.	−0.0495784
$596$	0	0
$597$	3.25849e6	0.374180
$598$	0	0
$599$	−15669.6	−0.00178439	−0.000892195	−	1.00000i	$-0.500284\pi$
$599$	−15669.6	−0.00178439	−0.000892195		1.00000i	$0.500284\pi$
$600$	0	0
$601$	−1.39658e7	−1.57718	−0.788590	−	0.614920i	$-0.789188\pi$
$601$	−1.39658e7	−1.57718	−0.788590	+	0.614920i	$0.789188\pi$
$602$	0	0
$603$	−2.22047e6	−0.248686
$604$	0	0
$605$	6.75844e6	0.750686
$606$	0	0
$607$	−3.09291e6	−0.340718	−0.170359	−	0.985382i	$-0.554493\pi$
$607$	−3.09291e6	−0.340718	−0.170359	+	0.985382i	$0.554493\pi$
$608$	0	0
$609$	−1.05297e6	−0.115046
$610$	0	0
$611$	−1.86247e6	−0.201830
$612$	0	0
$613$	1.31608e6	0.141459	0.0707296	−	0.997496i	$-0.477467\pi$
$613$	1.31608e6	0.141459	0.0707296	+	0.997496i	$0.477467\pi$
$614$	0	0
$615$	−1.52198e6	−0.162263
$616$	0	0
$617$	−1.33564e7	−1.41246	−0.706230	−	0.707983i	$-0.749605\pi$
$617$	−1.33564e7	−1.41246	−0.706230	+	0.707983i	$0.749605\pi$
$618$	0	0
$619$	−1.73558e7	−1.82062	−0.910308	−	0.413932i	$-0.864155\pi$
$619$	−1.73558e7	−1.82062	−0.910308	+	0.413932i	$0.864155\pi$
$620$	0	0
$621$	−1.48681e6	−0.154713
$622$	0	0
$623$	−2.26285e6	−0.233580
$624$	0	0
$625$	−5.94878e6	−0.609155
$626$	0	0
$627$	858577.	0.0872189
$628$	0	0
$629$	2.83489e6	0.285699
$630$	0	0
$631$	1.76228e6	0.176198	0.0880990	−	0.996112i	$-0.471921\pi$
$631$	1.76228e6	0.176198	0.0880990	+	0.996112i	$0.471921\pi$
$632$	0	0
$633$	2.76107e6	0.273885
$634$	0	0
$635$	−813626.	−0.0800738
$636$	0	0
$637$	1.33259e6	0.130122
$638$	0	0
$639$	1.35997e7	1.31758
$640$	0	0
$641$	1.65464e7	1.59059	0.795293	−	0.606225i	$-0.207317\pi$
$641$	1.65464e7	1.59059	0.795293	+	0.606225i	$0.207317\pi$
$642$	0	0
$643$	−4.94714e6	−0.471874	−0.235937	−	0.971768i	$-0.575816\pi$
$643$	−4.94714e6	−0.471874	−0.235937	+	0.971768i	$0.575816\pi$
$644$	0	0
$645$	2.29436e6	0.217151
$646$	0	0
$647$	1.44182e7	1.35410	0.677050	−	0.735937i	$-0.263258\pi$
$647$	1.44182e7	1.35410	0.677050	+	0.735937i	$0.263258\pi$
$648$	0	0
$649$	−636043.	−0.0592754
$650$	0	0
$651$	805460.	0.0744888
$652$	0	0
$653$	−5.14167e6	−0.471869	−0.235934	−	0.971769i	$-0.575815\pi$
$653$	−5.14167e6	−0.471869	−0.235934	+	0.971769i	$0.575815\pi$
$654$	0	0
$655$	−9.18492e6	−0.836512
$656$	0	0
$657$	−8.83467e6	−0.798504
$658$	0	0
$659$	−4.53101e6	−0.406426	−0.203213	−	0.979135i	$-0.565138\pi$
$659$	−4.53101e6	−0.406426	−0.203213	+	0.979135i	$0.565138\pi$
$660$	0	0
$661$	8.85220e6	0.788039	0.394019	−	0.919102i	$-0.371084\pi$
$661$	8.85220e6	0.788039	0.394019	+	0.919102i	$0.371084\pi$
$662$	0	0
$663$	152756.	0.0134963
$664$	0	0
$665$	−1.56785e6	−0.137484
$666$	0	0
$667$	−3.33567e6	−0.290314
$668$	0	0
$669$	−1.49713e6	−0.129329
$670$	0	0
$671$	849758.	0.0728600
$672$	0	0
$673$	−2.81577e6	−0.239640	−0.119820	−	0.992796i	$-0.538232\pi$
$673$	−2.81577e6	−0.239640	−0.119820	+	0.992796i	$0.538232\pi$
$674$	0	0
$675$	−2.50719e6	−0.211801
$676$	0	0
$677$	8.21225e6	0.688638	0.344319	−	0.938853i	$-0.388110\pi$
$677$	8.21225e6	0.688638	0.344319	+	0.938853i	$0.388110\pi$
$678$	0	0
$679$	4.25649e6	0.354305
$680$	0	0
$681$	7.54536e6	0.623465
$682$	0	0
$683$	2.09800e7	1.72090	0.860448	−	0.509538i	$-0.170184\pi$
$683$	2.09800e7	1.72090	0.860448	+	0.509538i	$0.170184\pi$
$684$	0	0
$685$	−4.46253e6	−0.363375
$686$	0	0
$687$	−3.68791e6	−0.298118
$688$	0	0
$689$	−806495.	−0.0647223
$690$	0	0
$691$	343027.	0.0273296	0.0136648	−	0.999907i	$-0.495650\pi$
$691$	343027.	0.0273296	0.0136648	+	0.999907i	$0.495650\pi$
$692$	0	0
$693$	793150.	0.0627368
$694$	0	0
$695$	337337.	0.0264912
$696$	0	0
$697$	−1.69566e6	−0.132207
$698$	0	0
$699$	−4.11525e6	−0.318569
$700$	0	0
$701$	−1.09435e7	−0.841124	−0.420562	−	0.907264i	$-0.638167\pi$
$701$	−1.09435e7	−0.841124	−0.420562	+	0.907264i	$0.638167\pi$
$702$	0	0
$703$	1.03814e7	0.792262
$704$	0	0
$705$	6.16035e6	0.466802
$706$	0	0
$707$	−98436.8	−0.00740643
$708$	0	0
$709$	−6.78987e6	−0.507278	−0.253639	−	0.967299i	$-0.581628\pi$
$709$	−6.78987e6	−0.507278	−0.253639	+	0.967299i	$0.581628\pi$
$710$	0	0
$711$	−4.25736e6	−0.315840
$712$	0	0
$713$	2.55159e6	0.187970
$714$	0	0
$715$	−502460.	−0.0367567
$716$	0	0
$717$	644016.	0.0467841
$718$	0	0
$719$	7.69216e6	0.554915	0.277457	−	0.960738i	$-0.410508\pi$
$719$	7.69216e6	0.554915	0.277457	+	0.960738i	$0.410508\pi$
$720$	0	0
$721$	−2.86407e6	−0.205185
$722$	0	0
$723$	6.01986e6	0.428293
$724$	0	0
$725$	−5.62491e6	−0.397439
$726$	0	0
$727$	−1.14796e7	−0.805547	−0.402774	−	0.915300i	$-0.631954\pi$
$727$	−1.14796e7	−0.805547	−0.402774	+	0.915300i	$0.631954\pi$
$728$	0	0
$729$	−3.38148e6	−0.235661
$730$	0	0
$731$	2.55618e6	0.176929
$732$	0	0
$733$	3.57719e6	0.245913	0.122957	−	0.992412i	$-0.460762\pi$
$733$	3.57719e6	0.245913	0.122957	+	0.992412i	$0.460762\pi$
$734$	0	0
$735$	−4.40771e6	−0.300950
$736$	0	0
$737$	−1.37137e6	−0.0930004
$738$	0	0
$739$	−1.73263e7	−1.16706	−0.583531	−	0.812091i	$-0.698329\pi$
$739$	−1.73263e7	−1.16706	−0.583531	+	0.812091i	$0.698329\pi$
$740$	0	0
$741$	559396.	0.0374261
$742$	0	0
$743$	1.47792e7	0.982154	0.491077	−	0.871116i	$-0.336603\pi$
$743$	1.47792e7	0.982154	0.491077	+	0.871116i	$0.336603\pi$
$744$	0	0
$745$	−1.97081e6	−0.130093
$746$	0	0
$747$	−2.36347e6	−0.154970
$748$	0	0
$749$	−3.24440e6	−0.211315
$750$	0	0
$751$	−1.55588e7	−1.00665	−0.503324	−	0.864098i	$-0.667890\pi$
$751$	−1.55588e7	−1.00665	−0.503324	+	0.864098i	$0.667890\pi$
$752$	0	0
$753$	3.19000e6	0.205024
$754$	0	0
$755$	2.48920e7	1.58925
$756$	0	0
$757$	5.43739e6	0.344866	0.172433	−	0.985021i	$-0.444837\pi$
$757$	5.43739e6	0.344866	0.172433	+	0.985021i	$0.444837\pi$
$758$	0	0
$759$	−423447.	−0.0266805
$760$	0	0
$761$	−1.27155e7	−0.795923	−0.397962	−	0.917402i	$-0.630282\pi$
$761$	−1.27155e7	−0.795923	−0.397962	+	0.917402i	$0.630282\pi$
$762$	0	0
$763$	1.40688e6	0.0874873
$764$	0	0
$765$	2.99805e6	0.185219
$766$	0	0
$767$	−414407.	−0.0254354
$768$	0	0
$769$	1.20062e7	0.732132	0.366066	−	0.930589i	$-0.380704\pi$
$769$	1.20062e7	0.732132	0.366066	+	0.930589i	$0.380704\pi$
$770$	0	0
$771$	−391010.	−0.0236893
$772$	0	0
$773$	1.57102e6	0.0945656	0.0472828	−	0.998882i	$-0.484944\pi$
$773$	1.57102e6	0.0945656	0.0472828	+	0.998882i	$0.484944\pi$
$774$	0	0
$775$	4.30273e6	0.257330
$776$	0	0
$777$	−1.61625e6	−0.0960409
$778$	0	0
$779$	−6.20953e6	−0.366619
$780$	0	0
$781$	8.39923e6	0.492733
$782$	0	0
$783$	1.59895e7	0.932029
$784$	0	0
$785$	−1.89948e7	−1.10017
$786$	0	0
$787$	4.28298e6	0.246496	0.123248	−	0.992376i	$-0.460669\pi$
$787$	4.28298e6	0.246496	0.123248	+	0.992376i	$0.460669\pi$
$788$	0	0
$789$	775172.	0.0443308
$790$	0	0
$791$	3.60646e6	0.204946
$792$	0	0
$793$	553651.	0.0312646
$794$	0	0
$795$	2.66758e6	0.149692
$796$	0	0
$797$	9.11725e6	0.508415	0.254207	−	0.967150i	$-0.418185\pi$
$797$	9.11725e6	0.508415	0.254207	+	0.967150i	$0.418185\pi$
$798$	0	0
$799$	6.86334e6	0.380337
$800$	0	0
$801$	1.58456e7	0.872624
$802$	0	0
$803$	−5.45632e6	−0.298614
$804$	0	0
$805$	773259.	0.0420567
$806$	0	0
$807$	−1.12582e7	−0.608537
$808$	0	0
$809$	−2.16543e7	−1.16325	−0.581625	−	0.813457i	$-0.697583\pi$
$809$	−2.16543e7	−1.16325	−0.581625	+	0.813457i	$0.697583\pi$
$810$	0	0
$811$	2.21451e7	1.18230	0.591148	−	0.806563i	$-0.298675\pi$
$811$	2.21451e7	1.18230	0.591148	+	0.806563i	$0.298675\pi$
$812$	0	0
$813$	2.04420e6	0.108467
$814$	0	0
$815$	−7.74602e6	−0.408493
$816$	0	0
$817$	9.36080e6	0.490634
$818$	0	0
$819$	516768.	0.0269207
$820$	0	0
$821$	−1.22016e7	−0.631771	−0.315885	−	0.948797i	$-0.602301\pi$
$821$	−1.22016e7	−0.631771	−0.315885	+	0.948797i	$0.602301\pi$
$822$	0	0
$823$	3.29978e7	1.69819	0.849094	−	0.528242i	$-0.177149\pi$
$823$	3.29978e7	1.69819	0.849094	+	0.528242i	$0.177149\pi$
$824$	0	0
$825$	−714055.	−0.0365255
$826$	0	0
$827$	1.73326e7	0.881253	0.440627	−	0.897690i	$-0.354756\pi$
$827$	1.73326e7	0.881253	0.440627	+	0.897690i	$0.354756\pi$
$828$	0	0
$829$	−2.11605e6	−0.106940	−0.0534700	−	0.998569i	$-0.517028\pi$
$829$	−2.11605e6	−0.106940	−0.0534700	+	0.998569i	$0.517028\pi$
$830$	0	0
$831$	5.50292e6	0.276434
$832$	0	0
$833$	−4.91070e6	−0.245206
$834$	0	0
$835$	1.76541e7	0.876255
$836$	0	0
$837$	−1.22310e7	−0.603461
$838$	0	0
$839$	−1.23948e7	−0.607904	−0.303952	−	0.952687i	$-0.598306\pi$
$839$	−1.23948e7	−0.607904	−0.303952	+	0.952687i	$0.598306\pi$
$840$	0	0
$841$	1.53613e7	0.748925
$842$	0	0
$843$	1.27550e7	0.618173
$844$	0	0
$845$	1.70317e7	0.820572
$846$	0	0
$847$	−4.29288e6	−0.205608
$848$	0	0
$849$	6.01756e6	0.286518
$850$	0	0
$851$	−5.12008e6	−0.242355
$852$	0	0
$853$	3.29452e6	0.155031	0.0775157	−	0.996991i	$-0.475301\pi$
$853$	3.29452e6	0.155031	0.0775157	+	0.996991i	$0.475301\pi$
$854$	0	0
$855$	1.09789e7	0.513623
$856$	0	0
$857$	−1.93524e7	−0.900081	−0.450041	−	0.893008i	$-0.648591\pi$
$857$	−1.93524e7	−0.900081	−0.450041	+	0.893008i	$0.648591\pi$
$858$	0	0
$859$	−2.22302e7	−1.02792	−0.513960	−	0.857814i	$-0.671822\pi$
$859$	−2.22302e7	−1.02792	−0.513960	+	0.857814i	$0.671822\pi$
$860$	0	0
$861$	966742.	0.0444429
$862$	0	0
$863$	1.82903e7	0.835978	0.417989	−	0.908452i	$-0.362735\pi$
$863$	1.82903e7	0.835978	0.417989	+	0.908452i	$0.362735\pi$
$864$	0	0
$865$	3.54775e7	1.61218
$866$	0	0
$867$	7.84263e6	0.354335
$868$	0	0
$869$	−2.62936e6	−0.118114
$870$	0	0
$871$	−893498.	−0.0399069
$872$	0	0
$873$	−2.98061e7	−1.32364
$874$	0	0
$875$	5.64278e6	0.249157
$876$	0	0
$877$	1.32774e7	0.582926	0.291463	−	0.956582i	$-0.405858\pi$
$877$	1.32774e7	0.582926	0.291463	+	0.956582i	$0.405858\pi$
$878$	0	0
$879$	3.88382e6	0.169546
$880$	0	0
$881$	2.14234e7	0.929928	0.464964	−	0.885330i	$-0.346067\pi$
$881$	2.14234e7	0.929928	0.464964	+	0.885330i	$0.346067\pi$
$882$	0	0
$883$	700573.	0.0302379	0.0151190	−	0.999886i	$-0.495187\pi$
$883$	700573.	0.0302379	0.0151190	+	0.999886i	$0.495187\pi$
$884$	0	0
$885$	1.37070e6	0.0588280
$886$	0	0
$887$	−4.68736e6	−0.200041	−0.100021	−	0.994985i	$-0.531891\pi$
$887$	−4.68736e6	−0.200041	−0.100021	+	0.994985i	$0.531891\pi$
$888$	0	0
$889$	516806.	0.0219317
$890$	0	0
$891$	−4.46027e6	−0.188221
$892$	0	0
$893$	2.51337e7	1.05470
$894$	0	0
$895$	−1.24663e7	−0.520213
$896$	0	0
$897$	−275892.	−0.0114487
$898$	0	0
$899$	−2.74404e7	−1.13238
$900$	0	0
$901$	2.97199e6	0.121965
$902$	0	0
$903$	−1.45735e6	−0.0594765
$904$	0	0
$905$	2.84189e7	1.15342
$906$	0	0
$907$	3.66031e7	1.47741	0.738703	−	0.674031i	$-0.235439\pi$
$907$	3.66031e7	1.47741	0.738703	+	0.674031i	$0.235439\pi$
$908$	0	0
$909$	689305.	0.0276695
$910$	0	0
$911$	1.32615e7	0.529414	0.264707	−	0.964329i	$-0.414725\pi$
$911$	1.32615e7	0.529414	0.264707	+	0.964329i	$0.414725\pi$
$912$	0	0
$913$	−1.45969e6	−0.0579539
$914$	0	0
$915$	−1.83127e6	−0.0723100
$916$	0	0
$917$	5.83416e6	0.229116
$918$	0	0
$919$	3.73882e7	1.46031	0.730156	−	0.683280i	$-0.239447\pi$
$919$	3.73882e7	1.46031	0.730156	+	0.683280i	$0.239447\pi$
$920$	0	0
$921$	−1.96475e6	−0.0763235
$922$	0	0
$923$	5.47243e6	0.211434
$924$	0	0
$925$	−8.63395e6	−0.331784
$926$	0	0
$927$	2.00557e7	0.766547
$928$	0	0
$929$	−2.76210e7	−1.05003	−0.525013	−	0.851094i	$-0.675940\pi$
$929$	−2.76210e7	−1.05003	−0.525013	+	0.851094i	$0.675940\pi$
$930$	0	0
$931$	−1.79831e7	−0.679971
$932$	0	0
$933$	−1.21256e7	−0.456036
$934$	0	0
$935$	1.85160e6	0.0692657
$936$	0	0
$937$	1.58687e7	0.590462	0.295231	−	0.955426i	$-0.404603\pi$
$937$	1.58687e7	0.590462	0.295231	+	0.955426i	$0.404603\pi$
$938$	0	0
$939$	−132909.	−0.00491915
$940$	0	0
$941$	−2.80899e7	−1.03413	−0.517066	−	0.855946i	$-0.672976\pi$
$941$	−2.80899e7	−1.03413	−0.517066	+	0.855946i	$0.672976\pi$
$942$	0	0
$943$	3.06252e6	0.112150
$944$	0	0
$945$	−3.70661e6	−0.135020
$946$	0	0
$947$	−1.05595e7	−0.382619	−0.191309	−	0.981530i	$-0.561273\pi$
$947$	−1.05595e7	−0.382619	−0.191309	+	0.981530i	$0.561273\pi$
$948$	0	0
$949$	−3.55500e6	−0.128137
$950$	0	0
$951$	−2.76418e6	−0.0991093
$952$	0	0
$953$	2.88692e7	1.02968	0.514840	−	0.857287i	$-0.327851\pi$
$953$	2.88692e7	1.02968	0.514840	+	0.857287i	$0.327851\pi$
$954$	0	0
$955$	257744.	0.00914492
$956$	0	0
$957$	4.55384e6	0.160730
$958$	0	0
$959$	2.83455e6	0.0995262
$960$	0	0
$961$	−7.63883e6	−0.266820
$962$	0	0
$963$	2.27189e7	0.789446
$964$	0	0
$965$	−2.38495e7	−0.824445
$966$	0	0
$967$	2.02367e7	0.695943	0.347971	−	0.937505i	$-0.386871\pi$
$967$	2.02367e7	0.695943	0.347971	+	0.937505i	$0.386871\pi$
$968$	0	0
$969$	−2.06141e6	−0.0705271
$970$	0	0
$971$	1.52327e7	0.518477	0.259238	−	0.965813i	$-0.416529\pi$
$971$	1.52327e7	0.518477	0.259238	+	0.965813i	$0.416529\pi$
$972$	0	0
$973$	−214272.	−0.00725578
$974$	0	0
$975$	−465234.	−0.0156733
$976$	0	0
$977$	4.61516e7	1.54686	0.773429	−	0.633883i	$-0.218540\pi$
$977$	4.61516e7	1.54686	0.773429	+	0.633883i	$0.218540\pi$
$978$	0	0
$979$	9.78628e6	0.326333
$980$	0	0
$981$	−9.85167e6	−0.326842
$982$	0	0
$983$	−1.88005e7	−0.620561	−0.310281	−	0.950645i	$-0.600423\pi$
$983$	−1.88005e7	−0.620561	−0.310281	+	0.950645i	$0.600423\pi$
$984$	0	0
$985$	8.70712e6	0.285946
$986$	0	0
$987$	−3.91298e6	−0.127854
$988$	0	0
$989$	−4.61671e6	−0.150086
$990$	0	0
$991$	−3.85747e7	−1.24772	−0.623862	−	0.781535i	$-0.714437\pi$
$991$	−3.85747e7	−1.24772	−0.623862	+	0.781535i	$0.714437\pi$
$992$	0	0
$993$	−1.33713e6	−0.0430330
$994$	0	0
$995$	2.57339e7	0.824038
$996$	0	0
$997$	−4.37506e7	−1.39395	−0.696973	−	0.717097i	$-0.745470\pi$
$997$	−4.37506e7	−1.39395	−0.696973	+	0.717097i	$0.745470\pi$
$998$	0	0
$999$	2.45430e7	0.778062

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1028.6.a.a.1.20	✓	49

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1028.6.a.a.1.20	✓	49	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:	\( N \)	\(=\)	\( 1028 = 2^{2} \cdot 257 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	1028.a (trivial)

\(f(q)\)	\(=\)	\(q-5.91999 q^{3} -46.7531 q^{5} +29.6970 q^{7} -207.954 q^{9} +O(q^{10})\)	\(q-5.91999 q^{3} -46.7531 q^{5} +29.6970 q^{7} -207.954 q^{9} -128.433 q^{11} -83.6789 q^{13} +276.778 q^{15} +308.362 q^{17} +1129.23 q^{19} -175.806 q^{21} -556.932 q^{23} -939.150 q^{25} +2669.64 q^{27} +5989.36 q^{29} -4581.52 q^{31} +760.321 q^{33} -1388.43 q^{35} +9193.37 q^{37} +495.379 q^{39} -5498.91 q^{41} +8289.54 q^{43} +9722.47 q^{45} +22257.4 q^{47} -15925.1 q^{49} -1825.50 q^{51} +9637.98 q^{53} +6004.62 q^{55} -6685.04 q^{57} +4952.35 q^{59} -6616.37 q^{61} -6175.61 q^{63} +3912.25 q^{65} +10677.7 q^{67} +3297.03 q^{69} -65397.9 q^{71} +42483.8 q^{73} +5559.76 q^{75} -3814.07 q^{77} +20472.6 q^{79} +34728.5 q^{81} +11365.4 q^{83} -14416.9 q^{85} -35457.0 q^{87} -76197.7 q^{89} -2485.01 q^{91} +27122.6 q^{93} -52795.0 q^{95} +143330. q^{97} +26708.1 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 49 q - 27 q^{3} - 89 q^{5} + 88 q^{7} + 3430 q^{9}+O(q^{10}) \) 49 * q - 27 * q^3 - 89 * q^5 + 88 * q^7 + 3430 * q^9	\( 49 q - 27 q^{3} - 89 q^{5} + 88 q^{7} + 3430 q^{9} - 484 q^{11} - 2922 q^{13} - 244 q^{15} - 3638 q^{17} - 3567 q^{19} - 3811 q^{21} + 3712 q^{23} + 13568 q^{25} - 8814 q^{27} - 5911 q^{29} - 3539 q^{31} - 17215 q^{33} - 10524 q^{35} - 31531 q^{37} - 14986 q^{39} - 35365 q^{41} - 44830 q^{43} - 34403 q^{45} + 12410 q^{47} + 68891 q^{49} - 48608 q^{51} - 79861 q^{53} - 34265 q^{55} - 109824 q^{57} - 51564 q^{59} - 66228 q^{61} + 24981 q^{63} - 31903 q^{65} - 80135 q^{67} + 50903 q^{69} + 162703 q^{71} - 58717 q^{73} + 207827 q^{75} + 89720 q^{77} + 125647 q^{79} + 420081 q^{81} - 28722 q^{83} - 125617 q^{85} + 65848 q^{87} + 93837 q^{89} - 66437 q^{91} - 278711 q^{93} - 83170 q^{95} - 347060 q^{97} + 7934 q^{99}+O(q^{100}) \) 49 * q - 27 * q^3 - 89 * q^5 + 88 * q^7 + 3430 * q^9 - 484 * q^11 - 2922 * q^13 - 244 * q^15 - 3638 * q^17 - 3567 * q^19 - 3811 * q^21 + 3712 * q^23 + 13568 * q^25 - 8814 * q^27 - 5911 * q^29 - 3539 * q^31 - 17215 * q^33 - 10524 * q^35 - 31531 * q^37 - 14986 * q^39 - 35365 * q^41 - 44830 * q^43 - 34403 * q^45 + 12410 * q^47 + 68891 * q^49 - 48608 * q^51 - 79861 * q^53 - 34265 * q^55 - 109824 * q^57 - 51564 * q^59 - 66228 * q^61 + 24981 * q^63 - 31903 * q^65 - 80135 * q^67 + 50903 * q^69 + 162703 * q^71 - 58717 * q^73 + 207827 * q^75 + 89720 * q^77 + 125647 * q^79 + 420081 * q^81 - 28722 * q^83 - 125617 * q^85 + 65848 * q^87 + 93837 * q^89 - 66437 * q^91 - 278711 * q^93 - 83170 * q^95 - 347060 * q^97 + 7934 * q^99

Introduction

L-functions

Modular forms

Varieties

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Groups

Database

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