LMFDB - Embedded newform 1028.6.a.a.1.16

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			1.16
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)

$f(q)$	$=$	$q-12.7314 q^{3} -83.6517 q^{5} +121.595 q^{7} -80.9122 q^{9} +O(q^{10})$	$q-12.7314 q^{3} -83.6517 q^{5} +121.595 q^{7} -80.9122 q^{9} +227.549 q^{11} -501.962 q^{13} +1065.00 q^{15} -1301.45 q^{17} +23.0317 q^{19} -1548.07 q^{21} -1193.18 q^{23} +3872.60 q^{25} +4123.85 q^{27} +4353.07 q^{29} +4118.99 q^{31} -2897.02 q^{33} -10171.6 q^{35} +4011.09 q^{37} +6390.67 q^{39} -893.948 q^{41} -13106.6 q^{43} +6768.44 q^{45} -18721.1 q^{47} -2021.70 q^{49} +16569.3 q^{51} +1531.13 q^{53} -19034.9 q^{55} -293.225 q^{57} +36261.2 q^{59} +46497.1 q^{61} -9838.51 q^{63} +41990.0 q^{65} -29743.6 q^{67} +15190.8 q^{69} +82378.0 q^{71} +54789.4 q^{73} -49303.5 q^{75} +27668.8 q^{77} -18272.3 q^{79} -32840.5 q^{81} +47018.1 q^{83} +108869. q^{85} -55420.6 q^{87} +120815. q^{89} -61036.0 q^{91} -52440.4 q^{93} -1926.64 q^{95} +35190.9 q^{97} -18411.5 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−12.7314	−0.816718	−0.408359	−	0.912821i	$-0.633899\pi$
$3$	−12.7314	−0.816718	−0.408359	+	0.912821i	$0.633899\pi$
$4$	0	0
$5$	−83.6517	−1.49641	−0.748203	−	0.663470i	$-0.769083\pi$
$5$	−83.6517	−1.49641	−0.748203	+	0.663470i	$0.769083\pi$
$6$	0	0
$7$	121.595	0.937929	0.468965	−	0.883217i	$-0.344627\pi$
$7$	121.595	0.937929	0.468965	+	0.883217i	$0.344627\pi$
$8$	0	0
$9$	−80.9122	−0.332972
$10$	0	0
$11$	227.549	0.567014	0.283507	−	0.958970i	$-0.408502\pi$
$11$	227.549	0.567014	0.283507	+	0.958970i	$0.408502\pi$
$12$	0	0
$13$	−501.962	−0.823783	−0.411891	−	0.911233i	$-0.635132\pi$
$13$	−501.962	−0.823783	−0.411891	+	0.911233i	$0.635132\pi$
$14$	0	0
$15$	1065.00	1.22214
$16$	0	0
$17$	−1301.45	−1.09221	−0.546106	−	0.837716i	$-0.683890\pi$
$17$	−1301.45	−1.09221	−0.546106	+	0.837716i	$0.683890\pi$
$18$	0	0
$19$	23.0317	0.0146366	0.00731832	−	0.999973i	$-0.497670\pi$
$19$	23.0317	0.0146366	0.00731832	+	0.999973i	$0.497670\pi$
$20$	0	0
$21$	−1548.07	−0.766023
$22$	0	0
$23$	−1193.18	−0.470311	−0.235155	−	0.971958i	$-0.575560\pi$
$23$	−1193.18	−0.470311	−0.235155	+	0.971958i	$0.575560\pi$
$24$	0	0
$25$	3872.60	1.23923
$26$	0	0
$27$	4123.85	1.08866
$28$	0	0
$29$	4353.07	0.961171	0.480586	−	0.876948i	$-0.340424\pi$
$29$	4353.07	0.961171	0.480586	+	0.876948i	$0.340424\pi$
$30$	0	0
$31$	4118.99	0.769815	0.384908	−	0.922955i	$-0.374233\pi$
$31$	4118.99	0.769815	0.384908	+	0.922955i	$0.374233\pi$
$32$	0	0
$33$	−2897.02	−0.463091
$34$	0	0
$35$	−10171.6	−1.40352
$36$	0	0
$37$	4011.09	0.481680	0.240840	−	0.970565i	$-0.422577\pi$
$37$	4011.09	0.481680	0.240840	+	0.970565i	$0.422577\pi$
$38$	0	0
$39$	6390.67	0.672798
$40$	0	0
$41$	−893.948	−0.0830525	−0.0415262	−	0.999137i	$-0.513222\pi$
$41$	−893.948	−0.0830525	−0.0415262	+	0.999137i	$0.513222\pi$
$42$	0	0
$43$	−13106.6	−1.08099	−0.540493	−	0.841348i	$-0.681762\pi$
$43$	−13106.6	−1.08099	−0.540493	+	0.841348i	$0.681762\pi$
$44$	0	0
$45$	6768.44	0.498262
$46$	0	0
$47$	−18721.1	−1.23619	−0.618097	−	0.786102i	$-0.712096\pi$
$47$	−18721.1	−1.23619	−0.618097	+	0.786102i	$0.712096\pi$
$48$	0	0
$49$	−2021.70	−0.120289
$50$	0	0
$51$	16569.3	0.892028
$52$	0	0
$53$	1531.13	0.0748726	0.0374363	−	0.999299i	$-0.488081\pi$
$53$	1531.13	0.0748726	0.0374363	+	0.999299i	$0.488081\pi$
$54$	0	0
$55$	−19034.9	−0.848484
$56$	0	0
$57$	−293.225	−0.0119540
$58$	0	0
$59$	36261.2	1.35616	0.678082	−	0.734986i	$-0.262812\pi$
$59$	36261.2	1.35616	0.678082	+	0.734986i	$0.262812\pi$
$60$	0	0
$61$	46497.1	1.59993	0.799966	−	0.600045i	$-0.204851\pi$
$61$	46497.1	1.59993	0.799966	+	0.600045i	$0.204851\pi$
$62$	0	0
$63$	−9838.51	−0.312304
$64$	0	0
$65$	41990.0	1.23271
$66$	0	0
$67$	−29743.6	−0.809481	−0.404741	−	0.914432i	$-0.632638\pi$
$67$	−29743.6	−0.809481	−0.404741	+	0.914432i	$0.632638\pi$
$68$	0	0
$69$	15190.8	0.384111
$70$	0	0
$71$	82378.0	1.93939	0.969695	−	0.244317i	$-0.0785638\pi$
$71$	82378.0	1.93939	0.969695	+	0.244317i	$0.0785638\pi$
$72$	0	0
$73$	54789.4	1.20334	0.601671	−	0.798744i	$-0.294502\pi$
$73$	54789.4	1.20334	0.601671	+	0.798744i	$0.294502\pi$
$74$	0	0
$75$	−49303.5	−1.01210
$76$	0	0
$77$	27668.8	0.531819
$78$	0	0
$79$	−18272.3	−0.329401	−0.164701	−	0.986344i	$-0.552666\pi$
$79$	−18272.3	−0.329401	−0.164701	+	0.986344i	$0.552666\pi$
$80$	0	0
$81$	−32840.5	−0.556157
$82$	0	0
$83$	47018.1	0.749151	0.374576	−	0.927196i	$-0.377788\pi$
$83$	47018.1	0.749151	0.374576	+	0.927196i	$0.377788\pi$
$84$	0	0
$85$	108869.	1.63439
$86$	0	0
$87$	−55420.6	−0.785006
$88$	0	0
$89$	120815.	1.61677	0.808383	−	0.588657i	$-0.200343\pi$
$89$	120815.	1.61677	0.808383	+	0.588657i	$0.200343\pi$
$90$	0	0
$91$	−61036.0	−0.772650
$92$	0	0
$93$	−52440.4	−0.628722
$94$	0	0
$95$	−1926.64	−0.0219024
$96$	0	0
$97$	35190.9	0.379753	0.189876	−	0.981808i	$-0.439191\pi$
$97$	35190.9	0.379753	0.189876	+	0.981808i	$0.439191\pi$
$98$	0	0
$99$	−18411.5	−0.188800
$100$	0	0
$101$	41316.0	0.403009	0.201505	−	0.979488i	$-0.435417\pi$
$101$	41316.0	0.403009	0.201505	+	0.979488i	$0.435417\pi$
$102$	0	0
$103$	−134790.	−1.25188	−0.625941	−	0.779870i	$-0.715285\pi$
$103$	−134790.	−1.25188	−0.625941	+	0.779870i	$0.715285\pi$
$104$	0	0
$105$	129499.	1.14628
$106$	0	0
$107$	−28997.9	−0.244854	−0.122427	−	0.992478i	$-0.539068\pi$
$107$	−28997.9	−0.244854	−0.122427	+	0.992478i	$0.539068\pi$
$108$	0	0
$109$	−132738.	−1.07011	−0.535054	−	0.844818i	$-0.679709\pi$
$109$	−132738.	−1.07011	−0.535054	+	0.844818i	$0.679709\pi$
$110$	0	0
$111$	−51066.7	−0.393396
$112$	0	0
$113$	79805.5	0.587945	0.293972	−	0.955814i	$-0.405023\pi$
$113$	79805.5	0.587945	0.293972	+	0.955814i	$0.405023\pi$
$114$	0	0
$115$	99811.2	0.703776
$116$	0	0
$117$	40614.9	0.274297
$118$	0	0
$119$	−158250.	−1.02442
$120$	0	0
$121$	−109272.	−0.678495
$122$	0	0
$123$	11381.2	0.0678304
$124$	0	0
$125$	−62538.2	−0.357989
$126$	0	0
$127$	29259.9	0.160977	0.0804885	−	0.996756i	$-0.474352\pi$
$127$	29259.9	0.160977	0.0804885	+	0.996756i	$0.474352\pi$
$128$	0	0
$129$	166865.	0.882861
$130$	0	0
$131$	156996.	0.799301	0.399651	−	0.916668i	$-0.369131\pi$
$131$	156996.	0.799301	0.399651	+	0.916668i	$0.369131\pi$
$132$	0	0
$133$	2800.53	0.0137281
$134$	0	0
$135$	−344967.	−1.62908
$136$	0	0
$137$	252087.	1.14749	0.573745	−	0.819034i	$-0.305490\pi$
$137$	252087.	1.14749	0.573745	+	0.819034i	$0.305490\pi$
$138$	0	0
$139$	118047.	0.518224	0.259112	−	0.965847i	$-0.416570\pi$
$139$	118047.	0.518224	0.259112	+	0.965847i	$0.416570\pi$
$140$	0	0
$141$	238345.	1.00962
$142$	0	0
$143$	−114221.	−0.467097
$144$	0	0
$145$	−364142.	−1.43830
$146$	0	0
$147$	25739.0	0.0982421
$148$	0	0
$149$	309717.	1.14288	0.571438	−	0.820645i	$-0.306386\pi$
$149$	309717.	1.14288	0.571438	+	0.820645i	$0.306386\pi$
$150$	0	0
$151$	493685.	1.76201	0.881003	−	0.473111i	$-0.156869\pi$
$151$	493685.	1.76201	0.881003	+	0.473111i	$0.156869\pi$
$152$	0	0
$153$	105304.	0.363676
$154$	0	0
$155$	−344560.	−1.15196
$156$	0	0
$157$	−582724.	−1.88675	−0.943374	−	0.331732i	$-0.892367\pi$
$157$	−582724.	−1.88675	−0.943374	+	0.331732i	$0.892367\pi$
$158$	0	0
$159$	−19493.4	−0.0611498
$160$	0	0
$161$	−145084.	−0.441118
$162$	0	0
$163$	−608289.	−1.79325	−0.896626	−	0.442789i	$-0.853989\pi$
$163$	−608289.	−1.79325	−0.896626	+	0.442789i	$0.853989\pi$
$164$	0	0
$165$	242340.	0.692972
$166$	0	0
$167$	−561515.	−1.55801	−0.779005	−	0.627018i	$-0.784275\pi$
$167$	−561515.	−1.55801	−0.779005	+	0.627018i	$0.784275\pi$
$168$	0	0
$169$	−119327.	−0.321382
$170$	0	0
$171$	−1863.54	−0.00487359
$172$	0	0
$173$	−171230.	−0.434976	−0.217488	−	0.976063i	$-0.569786\pi$
$173$	−171230.	−0.434976	−0.217488	+	0.976063i	$0.569786\pi$
$174$	0	0
$175$	470888.	1.16231
$176$	0	0
$177$	−461655.	−1.10760
$178$	0	0
$179$	−78514.5	−0.183154	−0.0915771	−	0.995798i	$-0.529191\pi$
$179$	−78514.5	−0.183154	−0.0915771	+	0.995798i	$0.529191\pi$
$180$	0	0
$181$	−150830.	−0.342208	−0.171104	−	0.985253i	$-0.554733\pi$
$181$	−150830.	−0.342208	−0.171104	+	0.985253i	$0.554733\pi$
$182$	0	0
$183$	−591972.	−1.30669
$184$	0	0
$185$	−335535.	−0.720789
$186$	0	0
$187$	−296145.	−0.619299
$188$	0	0
$189$	501438.	1.02109
$190$	0	0
$191$	−858834.	−1.70344	−0.851718	−	0.524000i	$-0.824439\pi$
$191$	−858834.	−1.70344	−0.851718	+	0.524000i	$0.824439\pi$
$192$	0	0
$193$	−352803.	−0.681772	−0.340886	−	0.940105i	$-0.610727\pi$
$193$	−352803.	−0.681772	−0.340886	+	0.940105i	$0.610727\pi$
$194$	0	0
$195$	−534590.	−1.00678
$196$	0	0
$197$	−1.07282e6	−1.96952	−0.984758	−	0.173930i	$-0.944353\pi$
$197$	−1.07282e6	−1.96952	−0.984758	+	0.173930i	$0.944353\pi$
$198$	0	0
$199$	344154.	0.616055	0.308028	−	0.951377i	$-0.400331\pi$
$199$	344154.	0.616055	0.308028	+	0.951377i	$0.400331\pi$
$200$	0	0
$201$	378677.	0.661118
$202$	0	0
$203$	529311.	0.901511
$204$	0	0
$205$	74780.2	0.124280
$206$	0	0
$207$	96542.6	0.156600
$208$	0	0
$209$	5240.84	0.00829918
$210$	0	0
$211$	−328202.	−0.507498	−0.253749	−	0.967270i	$-0.581664\pi$
$211$	−328202.	−0.507498	−0.253749	+	0.967270i	$0.581664\pi$
$212$	0	0
$213$	−1.04878e6	−1.58393
$214$	0	0
$215$	1.09639e6	1.61760
$216$	0	0
$217$	500848.	0.722032
$218$	0	0
$219$	−697543.	−0.982791
$220$	0	0
$221$	653281.	0.899745
$222$	0	0
$223$	−163203.	−0.219768	−0.109884	−	0.993944i	$-0.535048\pi$
$223$	−163203.	−0.219768	−0.109884	+	0.993944i	$0.535048\pi$
$224$	0	0
$225$	−313341.	−0.412630
$226$	0	0
$227$	−1.05671e6	−1.36110	−0.680552	−	0.732700i	$-0.738260\pi$
$227$	−1.05671e6	−1.36110	−0.680552	+	0.732700i	$0.738260\pi$
$228$	0	0
$229$	−811347.	−1.02239	−0.511197	−	0.859464i	$-0.670798\pi$
$229$	−811347.	−1.02239	−0.511197	+	0.859464i	$0.670798\pi$
$230$	0	0
$231$	−352262.	−0.434346
$232$	0	0
$233$	885585.	1.06866	0.534331	−	0.845275i	$-0.320563\pi$
$233$	885585.	1.06866	0.534331	+	0.845275i	$0.320563\pi$
$234$	0	0
$235$	1.56605e6	1.84985
$236$	0	0
$237$	232631.	0.269028
$238$	0	0
$239$	1.55974e6	1.76627	0.883135	−	0.469120i	$-0.155429\pi$
$239$	1.55974e6	1.76627	0.883135	+	0.469120i	$0.155429\pi$
$240$	0	0
$241$	8609.72	0.00954874	0.00477437	−	0.999989i	$-0.498480\pi$
$241$	8609.72	0.00954874	0.00477437	+	0.999989i	$0.498480\pi$
$242$	0	0
$243$	−583990.	−0.634438
$244$	0	0
$245$	169118.	0.180001
$246$	0	0
$247$	−11561.0	−0.0120574
$248$	0	0
$249$	−598604.	−0.611845
$250$	0	0
$251$	−1.53039e6	−1.53327	−0.766634	−	0.642084i	$-0.778070\pi$
$251$	−1.53039e6	−1.53327	−0.766634	+	0.642084i	$0.778070\pi$
$252$	0	0
$253$	−271507.	−0.266673
$254$	0	0
$255$	−1.38605e6	−1.33484
$256$	0	0
$257$	66049.0	0.0623783
$257$	66049.0	0.0623783
$258$	0	0
$259$	487728.	0.451781
$260$	0	0
$261$	−352217.	−0.320043
$262$	0	0
$263$	106636.	0.0950637	0.0475318	−	0.998870i	$-0.484864\pi$
$263$	106636.	0.0950637	0.0475318	+	0.998870i	$0.484864\pi$
$264$	0	0
$265$	−128082.	−0.112040
$266$	0	0
$267$	−1.53815e6	−1.32044
$268$	0	0
$269$	−1.16782e6	−0.983999	−0.492000	−	0.870595i	$-0.663734\pi$
$269$	−1.16782e6	−0.983999	−0.492000	+	0.870595i	$0.663734\pi$
$270$	0	0
$271$	238125.	0.196962	0.0984809	−	0.995139i	$-0.468602\pi$
$271$	238125.	0.196962	0.0984809	+	0.995139i	$0.468602\pi$
$272$	0	0
$273$	777072.	0.631037
$274$	0	0
$275$	881208.	0.702663
$276$	0	0
$277$	−1.33204e6	−1.04308	−0.521541	−	0.853226i	$-0.674643\pi$
$277$	−1.33204e6	−1.04308	−0.521541	+	0.853226i	$0.674643\pi$
$278$	0	0
$279$	−333277.	−0.256327
$280$	0	0
$281$	−174111.	−0.131541	−0.0657705	−	0.997835i	$-0.520951\pi$
$281$	−174111.	−0.131541	−0.0657705	+	0.997835i	$0.520951\pi$
$282$	0	0
$283$	1.45171e6	1.07749	0.538746	−	0.842468i	$-0.318898\pi$
$283$	1.45171e6	1.07749	0.538746	+	0.842468i	$0.318898\pi$
$284$	0	0
$285$	24528.7	0.0178880
$286$	0	0
$287$	−108699.	−0.0778973
$288$	0	0
$289$	273926.	0.192925
$290$	0	0
$291$	−448028.	−0.310151
$292$	0	0
$293$	2.42266e6	1.64863	0.824316	−	0.566131i	$-0.191560\pi$
$293$	2.42266e6	1.64863	0.824316	+	0.566131i	$0.191560\pi$
$294$	0	0
$295$	−3.03331e6	−2.02937
$296$	0	0
$297$	938379.	0.617287
$298$	0	0
$299$	598929.	0.387434
$300$	0	0
$301$	−1.59370e6	−1.01389
$302$	0	0
$303$	−526010.	−0.329145
$304$	0	0
$305$	−3.88956e6	−2.39415
$306$	0	0
$307$	822615.	0.498139	0.249069	−	0.968486i	$-0.419875\pi$
$307$	822615.	0.498139	0.249069	+	0.968486i	$0.419875\pi$
$308$	0	0
$309$	1.71606e6	1.02243
$310$	0	0
$311$	435436.	0.255284	0.127642	−	0.991820i	$-0.459259\pi$
$311$	435436.	0.255284	0.127642	+	0.991820i	$0.459259\pi$
$312$	0	0
$313$	1.89179e6	1.09147	0.545736	−	0.837957i	$-0.316250\pi$
$313$	1.89179e6	1.09147	0.545736	+	0.837957i	$0.316250\pi$
$314$	0	0
$315$	823008.	0.467334
$316$	0	0
$317$	−1.49632e6	−0.836329	−0.418164	−	0.908371i	$-0.637326\pi$
$317$	−1.49632e6	−0.836329	−0.418164	+	0.908371i	$0.637326\pi$
$318$	0	0
$319$	990539.	0.544998
$320$	0	0
$321$	369182.	0.199976
$322$	0	0
$323$	−29974.7	−0.0159863
$324$	0	0
$325$	−1.94390e6	−1.02086
$326$	0	0
$327$	1.68993e6	0.873976
$328$	0	0
$329$	−2.27639e6	−1.15946
$330$	0	0
$331$	−3.24944e6	−1.63019	−0.815095	−	0.579328i	$-0.803315\pi$
$331$	−3.24944e6	−1.63019	−0.815095	+	0.579328i	$0.803315\pi$
$332$	0	0
$333$	−324546.	−0.160386
$334$	0	0
$335$	2.48810e6	1.21131
$336$	0	0
$337$	4.00288e6	1.91998	0.959991	−	0.280029i	$-0.0903441\pi$
$337$	4.00288e6	1.91998	0.959991	+	0.280029i	$0.0903441\pi$
$338$	0	0
$339$	−1.01603e6	−0.480185
$340$	0	0
$341$	937274.	0.436496
$342$	0	0
$343$	−2.28947e6	−1.05075
$344$	0	0
$345$	−1.27073e6	−0.574787
$346$	0	0
$347$	−1.90315e6	−0.848497	−0.424248	−	0.905546i	$-0.639462\pi$
$347$	−1.90315e6	−0.848497	−0.424248	+	0.905546i	$0.639462\pi$
$348$	0	0
$349$	994424.	0.437027	0.218513	−	0.975834i	$-0.429879\pi$
$349$	994424.	0.437027	0.218513	+	0.975834i	$0.429879\pi$
$350$	0	0
$351$	−2.07002e6	−0.896821
$352$	0	0
$353$	2.44890e6	1.04600	0.523002	−	0.852331i	$-0.324812\pi$
$353$	2.44890e6	1.04600	0.523002	+	0.852331i	$0.324812\pi$
$354$	0	0
$355$	−6.89106e6	−2.90212
$356$	0	0
$357$	2.01474e6	0.836659
$358$	0	0
$359$	3.61643e6	1.48096	0.740481	−	0.672078i	$-0.234598\pi$
$359$	3.61643e6	1.48096	0.740481	+	0.672078i	$0.234598\pi$
$360$	0	0
$361$	−2.47557e6	−0.999786
$362$	0	0
$363$	1.39119e6	0.554139
$364$	0	0
$365$	−4.58322e6	−1.80069
$366$	0	0
$367$	−3.27846e6	−1.27059	−0.635294	−	0.772271i	$-0.719121\pi$
$367$	−3.27846e6	−1.27059	−0.635294	+	0.772271i	$0.719121\pi$
$368$	0	0
$369$	72331.3	0.0276542
$370$	0	0
$371$	186178.	0.0702252
$372$	0	0
$373$	254110.	0.0945691	0.0472845	−	0.998881i	$-0.484943\pi$
$373$	254110.	0.0945691	0.0472845	+	0.998881i	$0.484943\pi$
$374$	0	0
$375$	796196.	0.292376
$376$	0	0
$377$	−2.18508e6	−0.791797
$378$	0	0
$379$	−191783.	−0.0685823	−0.0342912	−	0.999412i	$-0.510917\pi$
$379$	−191783.	−0.0685823	−0.0342912	+	0.999412i	$0.510917\pi$
$380$	0	0
$381$	−372519.	−0.131473
$382$	0	0
$383$	−3.70863e6	−1.29186	−0.645932	−	0.763395i	$-0.723531\pi$
$383$	−3.70863e6	−1.29186	−0.645932	+	0.763395i	$0.723531\pi$
$384$	0	0
$385$	−2.31454e6	−0.795818
$386$	0	0
$387$	1.06049e6	0.359938
$388$	0	0
$389$	788062.	0.264050	0.132025	−	0.991246i	$-0.457852\pi$
$389$	788062.	0.264050	0.132025	+	0.991246i	$0.457852\pi$
$390$	0	0
$391$	1.55286e6	0.513679
$392$	0	0
$393$	−1.99877e6	−0.652803
$394$	0	0
$395$	1.52851e6	0.492918
$396$	0	0
$397$	−1.80408e6	−0.574487	−0.287243	−	0.957858i	$-0.592739\pi$
$397$	−1.80408e6	−0.574487	−0.287243	+	0.957858i	$0.592739\pi$
$398$	0	0
$399$	−35654.6	−0.0112120
$400$	0	0
$401$	−3.21508e6	−0.998462	−0.499231	−	0.866469i	$-0.666384\pi$
$401$	−3.21508e6	−0.998462	−0.499231	+	0.866469i	$0.666384\pi$
$402$	0	0
$403$	−2.06758e6	−0.634161
$404$	0	0
$405$	2.74717e6	0.832238
$406$	0	0
$407$	912722.	0.273119
$408$	0	0
$409$	−3.34056e6	−0.987441	−0.493720	−	0.869621i	$-0.664363\pi$
$409$	−3.34056e6	−0.987441	−0.493720	+	0.869621i	$0.664363\pi$
$410$	0	0
$411$	−3.20941e6	−0.937176
$412$	0	0
$413$	4.40917e6	1.27199
$414$	0	0
$415$	−3.93314e6	−1.12103
$416$	0	0
$417$	−1.50290e6	−0.423243
$418$	0	0
$419$	1.71964e6	0.478522	0.239261	−	0.970955i	$-0.423095\pi$
$419$	1.71964e6	0.478522	0.239261	+	0.970955i	$0.423095\pi$
$420$	0	0
$421$	5.90864e6	1.62473	0.812367	−	0.583146i	$-0.198179\pi$
$421$	5.90864e6	1.62473	0.812367	+	0.583146i	$0.198179\pi$
$422$	0	0
$423$	1.51477e6	0.411618
$424$	0	0
$425$	−5.04001e6	−1.35350
$426$	0	0
$427$	5.65381e6	1.50062
$428$	0	0
$429$	1.45419e6	0.381486
$430$	0	0
$431$	5.09141e6	1.32021	0.660107	−	0.751171i	$-0.270511\pi$
$431$	5.09141e6	1.32021	0.660107	+	0.751171i	$0.270511\pi$
$432$	0	0
$433$	−155245.	−0.0397922	−0.0198961	−	0.999802i	$-0.506334\pi$
$433$	−155245.	−0.0397922	−0.0198961	+	0.999802i	$0.506334\pi$
$434$	0	0
$435$	4.63602e6	1.17469
$436$	0	0
$437$	−27480.8	−0.00688377
$438$	0	0
$439$	2.68345e6	0.664556	0.332278	−	0.943181i	$-0.392183\pi$
$439$	2.68345e6	0.664556	0.332278	+	0.943181i	$0.392183\pi$
$440$	0	0
$441$	163580.	0.0400529
$442$	0	0
$443$	−4.61663e6	−1.11768	−0.558838	−	0.829277i	$-0.688753\pi$
$443$	−4.61663e6	−1.11768	−0.558838	+	0.829277i	$0.688753\pi$
$444$	0	0
$445$	−1.01064e7	−2.41934
$446$	0	0
$447$	−3.94312e6	−0.933407
$448$	0	0
$449$	−5.40374e6	−1.26497	−0.632483	−	0.774574i	$-0.717964\pi$
$449$	−5.40374e6	−1.26497	−0.632483	+	0.774574i	$0.717964\pi$
$450$	0	0
$451$	−203417.	−0.0470919
$452$	0	0
$453$	−6.28528e6	−1.43906
$454$	0	0
$455$	5.10577e6	1.15620
$456$	0	0
$457$	−63863.5	−0.0143042	−0.00715208	−	0.999974i	$-0.502277\pi$
$457$	−63863.5	−0.0143042	−0.00715208	+	0.999974i	$0.502277\pi$
$458$	0	0
$459$	−5.36700e6	−1.18905
$460$	0	0
$461$	−8.25381e6	−1.80885	−0.904424	−	0.426634i	$-0.859699\pi$
$461$	−8.25381e6	−1.80885	−0.904424	+	0.426634i	$0.859699\pi$
$462$	0	0
$463$	−4.67124e6	−1.01270	−0.506349	−	0.862329i	$-0.669005\pi$
$463$	−4.67124e6	−1.01270	−0.506349	+	0.862329i	$0.669005\pi$
$464$	0	0
$465$	4.38673e6	0.940823
$466$	0	0
$467$	124229.	0.0263591	0.0131796	−	0.999913i	$-0.495805\pi$
$467$	124229.	0.0263591	0.0131796	+	0.999913i	$0.495805\pi$
$468$	0	0
$469$	−3.61667e6	−0.759236
$470$	0	0
$471$	7.41887e6	1.54094
$472$	0	0
$473$	−2.98241e6	−0.612935
$474$	0	0
$475$	89192.5	0.0181382
$476$	0	0
$477$	−123887.	−0.0249305
$478$	0	0
$479$	1.47026e6	0.292790	0.146395	−	0.989226i	$-0.453233\pi$
$479$	1.47026e6	0.292790	0.146395	+	0.989226i	$0.453233\pi$
$480$	0	0
$481$	−2.01342e6	−0.396799
$482$	0	0
$483$	1.84712e6	0.360269
$484$	0	0
$485$	−2.94378e6	−0.568264
$486$	0	0
$487$	−7.16484e6	−1.36894	−0.684470	−	0.729041i	$-0.739966\pi$
$487$	−7.16484e6	−1.36894	−0.684470	+	0.729041i	$0.739966\pi$
$488$	0	0
$489$	7.74436e6	1.46458
$490$	0	0
$491$	4.14338e6	0.775623	0.387812	−	0.921739i	$-0.373231\pi$
$491$	4.14338e6	0.775623	0.387812	+	0.921739i	$0.373231\pi$
$492$	0	0
$493$	−5.66532e6	−1.04980
$494$	0	0
$495$	1.54016e6	0.282522
$496$	0	0
$497$	1.00167e7	1.81901
$498$	0	0
$499$	343909.	0.0618290	0.0309145	−	0.999522i	$-0.490158\pi$
$499$	343909.	0.0618290	0.0309145	+	0.999522i	$0.490158\pi$
$500$	0	0
$501$	7.14886e6	1.27245
$502$	0	0
$503$	8.68022e6	1.52972	0.764858	−	0.644199i	$-0.222809\pi$
$503$	8.68022e6	1.52972	0.764858	+	0.644199i	$0.222809\pi$
$504$	0	0
$505$	−3.45615e6	−0.603066
$506$	0	0
$507$	1.51919e6	0.262478
$508$	0	0
$509$	−1.85533e6	−0.317414	−0.158707	−	0.987326i	$-0.550733\pi$
$509$	−1.85533e6	−0.317414	−0.158707	+	0.987326i	$0.550733\pi$
$510$	0	0
$511$	6.66210e6	1.12865
$512$	0	0
$513$	94979.0	0.0159343
$514$	0	0
$515$	1.12754e7	1.87332
$516$	0	0
$517$	−4.25998e6	−0.700940
$518$	0	0
$519$	2.17999e6	0.355252
$520$	0	0
$521$	7.16632e6	1.15665	0.578325	−	0.815807i	$-0.303707\pi$
$521$	7.16632e6	1.15665	0.578325	+	0.815807i	$0.303707\pi$
$522$	0	0
$523$	8.43421e6	1.34831	0.674156	−	0.738589i	$-0.264508\pi$
$523$	8.43421e6	1.34831	0.674156	+	0.738589i	$0.264508\pi$
$524$	0	0
$525$	−5.99505e6	−0.949281
$526$	0	0
$527$	−5.36068e6	−0.840801
$528$	0	0
$529$	−5.01267e6	−0.778808
$530$	0	0
$531$	−2.93397e6	−0.451565
$532$	0	0
$533$	448728.	0.0684172
$534$	0	0
$535$	2.42572e6	0.366401
$536$	0	0
$537$	999597.	0.149585
$538$	0	0
$539$	−460036.	−0.0682055
$540$	0	0
$541$	−1.14824e7	−1.68670	−0.843350	−	0.537364i	$-0.819420\pi$
$541$	−1.14824e7	−1.68670	−0.843350	+	0.537364i	$0.819420\pi$
$542$	0	0
$543$	1.92027e6	0.279487
$544$	0	0
$545$	1.11037e7	1.60132
$546$	0	0
$547$	−477790.	−0.0682762	−0.0341381	−	0.999417i	$-0.510869\pi$
$547$	−477790.	−0.0682762	−0.0341381	+	0.999417i	$0.510869\pi$
$548$	0	0
$549$	−3.76219e6	−0.532733
$550$	0	0
$551$	100258.	0.0140683
$552$	0	0
$553$	−2.22181e6	−0.308955
$554$	0	0
$555$	4.27181e6	0.588681
$556$	0	0
$557$	5.93905e6	0.811109	0.405554	−	0.914071i	$-0.367078\pi$
$557$	5.93905e6	0.811109	0.405554	+	0.914071i	$0.367078\pi$
$558$	0	0
$559$	6.57904e6	0.890498
$560$	0	0
$561$	3.77033e6	0.505793
$562$	0	0
$563$	7.11742e6	0.946350	0.473175	−	0.880968i	$-0.343108\pi$
$563$	7.11742e6	0.946350	0.473175	+	0.880968i	$0.343108\pi$
$564$	0	0
$565$	−6.67586e6	−0.879804
$566$	0	0
$567$	−3.99324e6	−0.521636
$568$	0	0
$569$	1.99417e6	0.258215	0.129107	−	0.991631i	$-0.458789\pi$
$569$	1.99417e6	0.258215	0.129107	+	0.991631i	$0.458789\pi$
$570$	0	0
$571$	1.55890e6	0.200091	0.100046	−	0.994983i	$-0.468101\pi$
$571$	1.55890e6	0.200091	0.100046	+	0.994983i	$0.468101\pi$
$572$	0	0
$573$	1.09341e7	1.39123
$574$	0	0
$575$	−4.62070e6	−0.582825
$576$	0	0
$577$	3.98003e6	0.497676	0.248838	−	0.968545i	$-0.419951\pi$
$577$	3.98003e6	0.497676	0.248838	+	0.968545i	$0.419951\pi$
$578$	0	0
$579$	4.49166e6	0.556815
$580$	0	0
$581$	5.71715e6	0.702651
$582$	0	0
$583$	348408.	0.0424538
$584$	0	0
$585$	−3.39750e6	−0.410459
$586$	0	0
$587$	−1.08612e7	−1.30102	−0.650509	−	0.759499i	$-0.725444\pi$
$587$	−1.08612e7	−1.30102	−0.650509	+	0.759499i	$0.725444\pi$
$588$	0	0
$589$	94867.2	0.0112675
$590$	0	0
$591$	1.36584e7	1.60854
$592$	0	0
$593$	3.97783e6	0.464525	0.232263	−	0.972653i	$-0.425387\pi$
$593$	3.97783e6	0.464525	0.232263	+	0.972653i	$0.425387\pi$
$594$	0	0
$595$	1.32379e7	1.53294
$596$	0	0
$597$	−4.38155e6	−0.503143
$598$	0	0
$599$	2.00582e6	0.228416	0.114208	−	0.993457i	$-0.463567\pi$
$599$	2.00582e6	0.228416	0.114208	+	0.993457i	$0.463567\pi$
$600$	0	0
$601$	5.65004e6	0.638066	0.319033	−	0.947744i	$-0.396642\pi$
$601$	5.65004e6	0.638066	0.319033	+	0.947744i	$0.396642\pi$
$602$	0	0
$603$	2.40662e6	0.269535
$604$	0	0
$605$	9.14081e6	1.01530
$606$	0	0
$607$	8.19927e6	0.903241	0.451621	−	0.892210i	$-0.350846\pi$
$607$	8.19927e6	0.903241	0.451621	+	0.892210i	$0.350846\pi$
$608$	0	0
$609$	−6.73886e6	−0.736280
$610$	0	0
$611$	9.39729e6	1.01836
$612$	0	0
$613$	−5.75756e6	−0.618853	−0.309426	−	0.950923i	$-0.600137\pi$
$613$	−5.75756e6	−0.618853	−0.309426	+	0.950923i	$0.600137\pi$
$614$	0	0
$615$	−952055.	−0.101502
$616$	0	0
$617$	1.01155e7	1.06973	0.534867	−	0.844936i	$-0.320362\pi$
$617$	1.01155e7	1.06973	0.534867	+	0.844936i	$0.320362\pi$
$618$	0	0
$619$	−4.54969e6	−0.477260	−0.238630	−	0.971111i	$-0.576698\pi$
$619$	−4.54969e6	−0.477260	−0.238630	+	0.971111i	$0.576698\pi$
$620$	0	0
$621$	−4.92048e6	−0.512010
$622$	0	0
$623$	1.46905e7	1.51641
$624$	0	0
$625$	−6.87046e6	−0.703535
$626$	0	0
$627$	−66723.1	−0.00677809
$628$	0	0
$629$	−5.22025e6	−0.526096
$630$	0	0
$631$	−3.03282e6	−0.303231	−0.151616	−	0.988440i	$-0.548448\pi$
$631$	−3.03282e6	−0.303231	−0.151616	+	0.988440i	$0.548448\pi$
$632$	0	0
$633$	4.17845e6	0.414483
$634$	0	0
$635$	−2.44764e6	−0.240887
$636$	0	0
$637$	1.01481e6	0.0990919
$638$	0	0
$639$	−6.66539e6	−0.645763
$640$	0	0
$641$	2.48806e6	0.239175	0.119588	−	0.992824i	$-0.461843\pi$
$641$	2.48806e6	0.239175	0.119588	+	0.992824i	$0.461843\pi$
$642$	0	0
$643$	−6.59027e6	−0.628602	−0.314301	−	0.949323i	$-0.601770\pi$
$643$	−6.59027e6	−0.628602	−0.314301	+	0.949323i	$0.601770\pi$
$644$	0	0
$645$	−1.39586e7	−1.32112
$646$	0	0
$647$	1.63592e7	1.53639	0.768193	−	0.640219i	$-0.221156\pi$
$647$	1.63592e7	1.53639	0.768193	+	0.640219i	$0.221156\pi$
$648$	0	0
$649$	8.25121e6	0.768964
$650$	0	0
$651$	−6.37648e6	−0.589696
$652$	0	0
$653$	−1.00266e7	−0.920177	−0.460088	−	0.887873i	$-0.652182\pi$
$653$	−1.00266e7	−0.920177	−0.460088	+	0.887873i	$0.652182\pi$
$654$	0	0
$655$	−1.31330e7	−1.19608
$656$	0	0
$657$	−4.43313e6	−0.400679
$658$	0	0
$659$	9.77460e6	0.876770	0.438385	−	0.898787i	$-0.355551\pi$
$659$	9.77460e6	0.876770	0.438385	+	0.898787i	$0.355551\pi$
$660$	0	0
$661$	4.91810e6	0.437818	0.218909	−	0.975745i	$-0.429750\pi$
$661$	4.91810e6	0.437818	0.218909	+	0.975745i	$0.429750\pi$
$662$	0	0
$663$	−8.31716e6	−0.734838
$664$	0	0
$665$	−234269.	−0.0205429
$666$	0	0
$667$	−5.19398e6	−0.452049
$668$	0	0
$669$	2.07779e6	0.179489
$670$	0	0
$671$	1.05804e7	0.907184
$672$	0	0
$673$	−1.98757e6	−0.169155	−0.0845776	−	0.996417i	$-0.526954\pi$
$673$	−1.98757e6	−0.169155	−0.0845776	+	0.996417i	$0.526954\pi$
$674$	0	0
$675$	1.59700e7	1.34911
$676$	0	0
$677$	−1.50684e7	−1.26356	−0.631781	−	0.775147i	$-0.717676\pi$
$677$	−1.50684e7	−1.26356	−0.631781	+	0.775147i	$0.717676\pi$
$678$	0	0
$679$	4.27903e6	0.356181
$680$	0	0
$681$	1.34534e7	1.11164
$682$	0	0
$683$	−9.24134e6	−0.758024	−0.379012	−	0.925392i	$-0.623736\pi$
$683$	−9.24134e6	−0.758024	−0.379012	+	0.925392i	$0.623736\pi$
$684$	0	0
$685$	−2.10875e7	−1.71711
$686$	0	0
$687$	1.03296e7	0.835007
$688$	0	0
$689$	−768571.	−0.0616788
$690$	0	0
$691$	−1.80428e7	−1.43750	−0.718752	−	0.695267i	$-0.755286\pi$
$691$	−1.80428e7	−1.43750	−0.718752	+	0.695267i	$0.755286\pi$
$692$	0	0
$693$	−2.23875e6	−0.177081
$694$	0	0
$695$	−9.87482e6	−0.775474
$696$	0	0
$697$	1.16343e6	0.0907108
$698$	0	0
$699$	−1.12747e7	−0.872796
$700$	0	0
$701$	−2.00841e7	−1.54368	−0.771839	−	0.635818i	$-0.780663\pi$
$701$	−2.00841e7	−1.54368	−0.771839	+	0.635818i	$0.780663\pi$
$702$	0	0
$703$	92382.1	0.00705017
$704$	0	0
$705$	−1.99380e7	−1.51081
$706$	0	0
$707$	5.02382e6	0.377994
$708$	0	0
$709$	9.45437e6	0.706345	0.353173	−	0.935558i	$-0.385103\pi$
$709$	9.45437e6	0.706345	0.353173	+	0.935558i	$0.385103\pi$
$710$	0	0
$711$	1.47845e6	0.109681
$712$	0	0
$713$	−4.91468e6	−0.362052
$714$	0	0
$715$	9.55480e6	0.698966
$716$	0	0
$717$	−1.98576e7	−1.44254
$718$	0	0
$719$	−1.00206e6	−0.0722888	−0.0361444	−	0.999347i	$-0.511508\pi$
$719$	−1.00206e6	−0.0722888	−0.0361444	+	0.999347i	$0.511508\pi$
$720$	0	0
$721$	−1.63897e7	−1.17418
$722$	0	0
$723$	−109613.	−0.00779863
$724$	0	0
$725$	1.68577e7	1.19112
$726$	0	0
$727$	−8.69635e6	−0.610240	−0.305120	−	0.952314i	$-0.598697\pi$
$727$	−8.69635e6	−0.610240	−0.305120	+	0.952314i	$0.598697\pi$
$728$	0	0
$729$	1.54152e7	1.07431
$730$	0	0
$731$	1.70577e7	1.18067
$732$	0	0
$733$	1.91901e7	1.31922	0.659610	−	0.751608i	$-0.270721\pi$
$733$	1.91901e7	1.31922	0.659610	+	0.751608i	$0.270721\pi$
$734$	0	0
$735$	−2.15311e6	−0.147010
$736$	0	0
$737$	−6.76814e6	−0.458987
$738$	0	0
$739$	−1.58267e7	−1.06605	−0.533026	−	0.846099i	$-0.678945\pi$
$739$	−1.58267e7	−1.06605	−0.533026	+	0.846099i	$0.678945\pi$
$740$	0	0
$741$	147188.	0.00984750
$742$	0	0
$743$	−2.79443e7	−1.85704	−0.928520	−	0.371282i	$-0.878918\pi$
$743$	−2.79443e7	−1.85704	−0.928520	+	0.371282i	$0.878918\pi$
$744$	0	0
$745$	−2.59083e7	−1.71021
$746$	0	0
$747$	−3.80434e6	−0.249446
$748$	0	0
$749$	−3.52599e6	−0.229655
$750$	0	0
$751$	8.46082e6	0.547410	0.273705	−	0.961814i	$-0.411751\pi$
$751$	8.46082e6	0.547410	0.273705	+	0.961814i	$0.411751\pi$
$752$	0	0
$753$	1.94840e7	1.25225
$754$	0	0
$755$	−4.12976e7	−2.63668
$756$	0	0
$757$	−1.50915e7	−0.957178	−0.478589	−	0.878039i	$-0.658852\pi$
$757$	−1.50915e7	−0.957178	−0.478589	+	0.878039i	$0.658852\pi$
$758$	0	0
$759$	3.45665e6	0.217797
$760$	0	0
$761$	1.37882e7	0.863067	0.431533	−	0.902097i	$-0.357973\pi$
$761$	1.37882e7	0.863067	0.431533	+	0.902097i	$0.357973\pi$
$762$	0	0
$763$	−1.61402e7	−1.00369
$764$	0	0
$765$	−8.80882e6	−0.544207
$766$	0	0
$767$	−1.82018e7	−1.11718
$768$	0	0
$769$	−5.66143e6	−0.345232	−0.172616	−	0.984989i	$-0.555222\pi$
$769$	−5.66143e6	−0.345232	−0.172616	+	0.984989i	$0.555222\pi$
$770$	0	0
$771$	−840894.	−0.0509455
$772$	0	0
$773$	3.05717e7	1.84022	0.920111	−	0.391657i	$-0.128098\pi$
$773$	3.05717e7	1.84022	0.920111	+	0.391657i	$0.128098\pi$
$774$	0	0
$775$	1.59512e7	0.953980
$776$	0	0
$777$	−6.20945e6	−0.368978
$778$	0	0
$779$	−20589.1	−0.00121561
$780$	0	0
$781$	1.87451e7	1.09966
$782$	0	0
$783$	1.79514e7	1.04639
$784$	0	0
$785$	4.87458e7	2.82334
$786$	0	0
$787$	1.85272e7	1.06629	0.533143	−	0.846025i	$-0.321011\pi$
$787$	1.85272e7	1.06629	0.533143	+	0.846025i	$0.321011\pi$
$788$	0	0
$789$	−1.35762e6	−0.0776402
$790$	0	0
$791$	9.70393e6	0.551451
$792$	0	0
$793$	−2.33398e7	−1.31800
$794$	0	0
$795$	1.63066e6	0.0915049
$796$	0	0
$797$	1.53157e7	0.854068	0.427034	−	0.904236i	$-0.359558\pi$
$797$	1.53157e7	0.854068	0.427034	+	0.904236i	$0.359558\pi$
$798$	0	0
$799$	2.43647e7	1.35019
$800$	0	0
$801$	−9.77544e6	−0.538338
$802$	0	0
$803$	1.24673e7	0.682312
$804$	0	0
$805$	1.21365e7	0.660092
$806$	0	0
$807$	1.48679e7	0.803650
$808$	0	0
$809$	2.03022e7	1.09062	0.545308	−	0.838236i	$-0.316413\pi$
$809$	2.03022e7	1.09062	0.545308	+	0.838236i	$0.316413\pi$
$810$	0	0
$811$	−3.11561e7	−1.66338	−0.831690	−	0.555240i	$-0.812626\pi$
$811$	−3.11561e7	−1.66338	−0.831690	+	0.555240i	$0.812626\pi$
$812$	0	0
$813$	−3.03166e6	−0.160862
$814$	0	0
$815$	5.08844e7	2.68343
$816$	0	0
$817$	−301868.	−0.0158220
$818$	0	0
$819$	4.93856e6	0.257271
$820$	0	0
$821$	−3.04359e6	−0.157590	−0.0787950	−	0.996891i	$-0.525107\pi$
$821$	−3.04359e6	−0.157590	−0.0787950	+	0.996891i	$0.525107\pi$
$822$	0	0
$823$	3.07254e7	1.58124	0.790620	−	0.612307i	$-0.209758\pi$
$823$	3.07254e7	1.58124	0.790620	+	0.612307i	$0.209758\pi$
$824$	0	0
$825$	−1.12190e7	−0.573877
$826$	0	0
$827$	1.23000e6	0.0625377	0.0312688	−	0.999511i	$-0.490045\pi$
$827$	1.23000e6	0.0625377	0.0312688	+	0.999511i	$0.490045\pi$
$828$	0	0
$829$	−4.99198e6	−0.252282	−0.126141	−	0.992012i	$-0.540259\pi$
$829$	−4.99198e6	−0.252282	−0.126141	+	0.992012i	$0.540259\pi$
$830$	0	0
$831$	1.69587e7	0.851904
$832$	0	0
$833$	2.63114e6	0.131381
$834$	0	0
$835$	4.69717e7	2.33142
$836$	0	0
$837$	1.69861e7	0.838069
$838$	0	0
$839$	1.88310e7	0.923566	0.461783	−	0.886993i	$-0.347210\pi$
$839$	1.88310e7	0.923566	0.461783	+	0.886993i	$0.347210\pi$
$840$	0	0
$841$	−1.56191e6	−0.0761494
$842$	0	0
$843$	2.21668e6	0.107432
$844$	0	0
$845$	9.98189e6	0.480918
$846$	0	0
$847$	−1.32869e7	−0.636380
$848$	0	0
$849$	−1.84823e7	−0.880007
$850$	0	0
$851$	−4.78594e6	−0.226539
$852$	0	0
$853$	3.54279e7	1.66714	0.833571	−	0.552413i	$-0.186293\pi$
$853$	3.54279e7	1.66714	0.833571	+	0.552413i	$0.186293\pi$
$854$	0	0
$855$	155889.	0.00729288
$856$	0	0
$857$	−3.07480e7	−1.43010	−0.715048	−	0.699076i	$-0.753595\pi$
$857$	−3.07480e7	−1.43010	−0.715048	+	0.699076i	$0.753595\pi$
$858$	0	0
$859$	1.79634e7	0.830628	0.415314	−	0.909678i	$-0.363672\pi$
$859$	1.79634e7	0.830628	0.415314	+	0.909678i	$0.363672\pi$
$860$	0	0
$861$	1.38389e6	0.0636201
$862$	0	0
$863$	−1.77669e7	−0.812052	−0.406026	−	0.913862i	$-0.633086\pi$
$863$	−1.77669e7	−0.812052	−0.406026	+	0.913862i	$0.633086\pi$
$864$	0	0
$865$	1.43237e7	0.650900
$866$	0	0
$867$	−3.48746e6	−0.157566
$868$	0	0
$869$	−4.15785e6	−0.186775
$870$	0	0
$871$	1.49302e7	0.666837
$872$	0	0
$873$	−2.84737e6	−0.126447
$874$	0	0
$875$	−7.60432e6	−0.335769
$876$	0	0
$877$	6.18027e6	0.271337	0.135668	−	0.990754i	$-0.456682\pi$
$877$	6.18027e6	0.271337	0.135668	+	0.990754i	$0.456682\pi$
$878$	0	0
$879$	−3.08438e7	−1.34647
$880$	0	0
$881$	−1.24117e7	−0.538755	−0.269378	−	0.963035i	$-0.586818\pi$
$881$	−1.24117e7	−0.538755	−0.269378	+	0.963035i	$0.586818\pi$
$882$	0	0
$883$	1.32628e6	0.0572443	0.0286222	−	0.999590i	$-0.490888\pi$
$883$	1.32628e6	0.0572443	0.0286222	+	0.999590i	$0.490888\pi$
$884$	0	0
$885$	3.86182e7	1.65742
$886$	0	0
$887$	−8.91653e6	−0.380528	−0.190264	−	0.981733i	$-0.560934\pi$
$887$	−8.91653e6	−0.380528	−0.190264	+	0.981733i	$0.560934\pi$
$888$	0	0
$889$	3.55786e6	0.150985
$890$	0	0
$891$	−7.47285e6	−0.315349
$892$	0	0
$893$	−431178.	−0.0180937
$894$	0	0
$895$	6.56787e6	0.274073
$896$	0	0
$897$	−7.62519e6	−0.316424
$898$	0	0
$899$	1.79303e7	0.739924
$900$	0	0
$901$	−1.99270e6	−0.0817767
$902$	0	0
$903$	2.02900e7	0.828061
$904$	0	0
$905$	1.26172e7	0.512083
$906$	0	0
$907$	−2.84992e7	−1.15031	−0.575154	−	0.818045i	$-0.695058\pi$
$907$	−2.84992e7	−1.15031	−0.575154	+	0.818045i	$0.695058\pi$
$908$	0	0
$909$	−3.34297e6	−0.134191
$910$	0	0
$911$	−1.95509e7	−0.780496	−0.390248	−	0.920710i	$-0.627611\pi$
$911$	−1.95509e7	−0.780496	−0.390248	+	0.920710i	$0.627611\pi$
$912$	0	0
$913$	1.06989e7	0.424779
$914$	0	0
$915$	4.95194e7	1.95534
$916$	0	0
$917$	1.90899e7	0.749688
$918$	0	0
$919$	−4.25357e7	−1.66136	−0.830682	−	0.556748i	$-0.812049\pi$
$919$	−4.25357e7	−1.66136	−0.830682	+	0.556748i	$0.812049\pi$
$920$	0	0
$921$	−1.04730e7	−0.406839
$922$	0	0
$923$	−4.13506e7	−1.59764
$924$	0	0
$925$	1.55334e7	0.596913
$926$	0	0
$927$	1.09061e7	0.416842
$928$	0	0
$929$	−2.81545e7	−1.07031	−0.535155	−	0.844754i	$-0.679747\pi$
$929$	−2.81545e7	−1.07031	−0.535155	+	0.844754i	$0.679747\pi$
$930$	0	0
$931$	−46563.0	−0.00176062
$932$	0	0
$933$	−5.54370e6	−0.208495
$934$	0	0
$935$	2.47730e7	0.926724
$936$	0	0
$937$	1.04152e7	0.387543	0.193771	−	0.981047i	$-0.437928\pi$
$937$	1.04152e7	0.387543	0.193771	+	0.981047i	$0.437928\pi$
$938$	0	0
$939$	−2.40851e7	−0.891424
$940$	0	0
$941$	3.54348e7	1.30454	0.652268	−	0.757989i	$-0.273818\pi$
$941$	3.54348e7	1.30454	0.652268	+	0.757989i	$0.273818\pi$
$942$	0	0
$943$	1.06664e6	0.0390605
$944$	0	0
$945$	−4.19462e7	−1.52796
$946$	0	0
$947$	−1.41188e7	−0.511592	−0.255796	−	0.966731i	$-0.582338\pi$
$947$	−1.41188e7	−0.511592	−0.255796	+	0.966731i	$0.582338\pi$
$948$	0	0
$949$	−2.75022e7	−0.991292
$950$	0	0
$951$	1.90502e7	0.683044
$952$	0	0
$953$	1.53653e6	0.0548036	0.0274018	−	0.999625i	$-0.491277\pi$
$953$	1.53653e6	0.0548036	0.0274018	+	0.999625i	$0.491277\pi$
$954$	0	0
$955$	7.18429e7	2.54903
$956$	0	0
$957$	−1.26109e7	−0.445110
$958$	0	0
$959$	3.06525e7	1.07627
$960$	0	0
$961$	−1.16631e7	−0.407384
$962$	0	0
$963$	2.34628e6	0.0815294
$964$	0	0
$965$	2.95126e7	1.02021
$966$	0	0
$967$	1.77544e7	0.610575	0.305287	−	0.952260i	$-0.401247\pi$
$967$	1.77544e7	0.610575	0.305287	+	0.952260i	$0.401247\pi$
$968$	0	0
$969$	381618.	0.0130563
$970$	0	0
$971$	−3.92082e7	−1.33453	−0.667266	−	0.744820i	$-0.732535\pi$
$971$	−3.92082e7	−1.33453	−0.667266	+	0.744820i	$0.732535\pi$
$972$	0	0
$973$	1.43539e7	0.486058
$974$	0	0
$975$	2.47485e7	0.833753
$976$	0	0
$977$	−4.07391e7	−1.36545	−0.682724	−	0.730676i	$-0.739205\pi$
$977$	−4.07391e7	−1.36545	−0.682724	+	0.730676i	$0.739205\pi$
$978$	0	0
$979$	2.74915e7	0.916730
$980$	0	0
$981$	1.07401e7	0.356316
$982$	0	0
$983$	−1.82934e6	−0.0603826	−0.0301913	−	0.999544i	$-0.509612\pi$
$983$	−1.82934e6	−0.0603826	−0.0301913	+	0.999544i	$0.509612\pi$
$984$	0	0
$985$	8.97428e7	2.94720
$986$	0	0
$987$	2.89816e7	0.946954
$988$	0	0
$989$	1.56385e7	0.508400
$990$	0	0
$991$	1.87358e7	0.606023	0.303012	−	0.952987i	$-0.402008\pi$
$991$	1.87358e7	0.606023	0.303012	+	0.952987i	$0.402008\pi$
$992$	0	0
$993$	4.13698e7	1.33140
$994$	0	0
$995$	−2.87890e7	−0.921869
$996$	0	0
$997$	1.40301e7	0.447015	0.223508	−	0.974702i	$-0.428249\pi$
$997$	1.40301e7	0.447015	0.223508	+	0.974702i	$0.428249\pi$
$998$	0	0
$999$	1.65411e7	0.524386

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1028.6.a.a.1.16	✓	49

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1028.6.a.a.1.16	✓	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)

\(f(q)\)	\(=\)	\(q-12.7314 q^{3} -83.6517 q^{5} +121.595 q^{7} -80.9122 q^{9} +O(q^{10})\)	\(q-12.7314 q^{3} -83.6517 q^{5} +121.595 q^{7} -80.9122 q^{9} +227.549 q^{11} -501.962 q^{13} +1065.00 q^{15} -1301.45 q^{17} +23.0317 q^{19} -1548.07 q^{21} -1193.18 q^{23} +3872.60 q^{25} +4123.85 q^{27} +4353.07 q^{29} +4118.99 q^{31} -2897.02 q^{33} -10171.6 q^{35} +4011.09 q^{37} +6390.67 q^{39} -893.948 q^{41} -13106.6 q^{43} +6768.44 q^{45} -18721.1 q^{47} -2021.70 q^{49} +16569.3 q^{51} +1531.13 q^{53} -19034.9 q^{55} -293.225 q^{57} +36261.2 q^{59} +46497.1 q^{61} -9838.51 q^{63} +41990.0 q^{65} -29743.6 q^{67} +15190.8 q^{69} +82378.0 q^{71} +54789.4 q^{73} -49303.5 q^{75} +27668.8 q^{77} -18272.3 q^{79} -32840.5 q^{81} +47018.1 q^{83} +108869. q^{85} -55420.6 q^{87} +120815. q^{89} -61036.0 q^{91} -52440.4 q^{93} -1926.64 q^{95} +35190.9 q^{97} -18411.5 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 49 q - 27 q^{3} - 89 q^{5} + 88 q^{7} + 3430 q^{9}+O(q^{10}) \) 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

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more